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students truly understand the mathematical concepts, it's magic. Students who use this text are motivated to learn mathematics. They become more ...Show synopsisWhen students truly understand the mathematical concepts, it's magic. Students;from the textbook, to the eManipulative activities, to the online problem-solving tools and the resource-rich website;work in harmony to help achieve this
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Starting at $107.972History of Mathematics, A: An Introduction
Summary
One of the leading historians in the mathematics field, Victor Katz provides a world view of mathematics, balancing ancient, early modern, and modern history.
Author Biography
Victor J. Katz received his PhD in mathematics from Brandeis University in 1968 and has been Professor of Mathematics at the University of the District of Columbia for many years. He has long been interested in the history of mathematics and, in particular, in its use in teaching. He is the editor of The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook (2007). He has edited or co-edited two recent books dealing with this subject, Learn from the Masters (1994) and Using History to Teach Mathematics (2000). Dr. Katz also co-edited a collection of historical articles taken from MAA journals of the past 90 years, Sherlock Holmes in Babylon and other Tales of Mathematical History. He has directed two NSF-sponsored projects to help college teachers learn the history of mathematics and learn to use it in teaching. Dr. Katz has also involved secondary school teachers in writing materials using history in the teaching of various topics in the high school curriculum. These materials, Historical Modules for the Teaching and Learning of Mathematics, have now been published by the MAA. Currently, Dr. Katz is the PI on an NSF grant to the MAA that supports Convergence, an online magazine devoted to the history of mathematics and its use in teaching.
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Ideal for primary and secondary school teachers, Mathematica for the Classroom gives you the power to show animated classroom demos, create notebooks with quizzes and assignments, and quickly solve equations and create 3D graphs.
Schools that wish to license Mathematica for a group of teachers, a computer lab, or the entire campus can choose from a number of flexible and cost-effective ways to license, deploy, and administer Mathematica, many of which include benefits such as:
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This is a free, online textbook offered by BookBoon. "This is an introduction to linear algebra. The main part of the book...
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This is a free, online textbook offered by BookBoon. " This is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra.״
his text provides the reader with a solid foundation of the fundamental operations and concepts of matrix algebra. The topics...
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his text provides the reader with a solid foundation of the fundamental operations and concepts of matrix algebra. The topics include systems of linear equations, matrix arithmetic, transpose, trace, determinant, eigenvalues/vectors, and linear transformations, focusing largely on transformations of the Cartesian plane.The text is designed to be easily read, written in a casual style. Key concepts are explained, but rigorous proofs are omitted. Numerous examples are provided to illuminate new ideas and provide practice. Each section ends with exercises (with answers to odd questions appearing in the back).The text is currently in use by Cadets of the Virginia Military Institute. It is appropriate for undergraduates in mathematics and the sciences, as well as advanced high school students activity would be done at the end of the school year in a pre-algebra class. It is a way to introduce algebra and its...
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This activity would be done at the end of the school year in a pre-algebra class. It is a way to introduce algebra and its history, putting some personality into the abstractness of the subject by researching the individuals behind algebraic concepts. It was initially found on the following site five years ago when I first did it with my classes: It has since disappeared, however, so the specific modifications I made at the time are fuzzy at best, but I have made recent adjustments to every portion.Introduction:Algebra, what does it mean? Where did it come from? Who thought up this stuff? Have you ever wondered what the word algebra means or when and where algebra was developed or who developed algebraic concepts? In this project your group will go on a journey through time and the history of mathematics to discover the answers to these questions.Task:Each group will go on a quest to find the mathematicians' histories that have named as being the fathers or founders of algebra. On this journey your group will collect information about the mathematician responsible for developing the algebraic concept assigned to your group, create a timeline to show when the concept was developed in relation to other significant events in history, and find examples of the algebraic concept. Each group will prepare a Powerpoint to present the information to the class.Group I The Father of Algebra (Algebraic thought and equations)Group II Founder of Cartesian Plane and Graphing EquationsGroup III Developer of PolynomialsGroup IV Set Notation and Venn Diagrams DesignerEach group will need a Researcher, Recorder, Mathematician, and a Reporter.Researcher - Using the resources below, work with the Recorder to find and record needed information for your topic.Recorder - Record information on your topic and citation for where the information was found. Work with the Researcher and the Reporter to prepare a report of the findings of your group.Mathematician - Work with the Researcher and the Recorder to find examples of mathematical problems from your assigned topic. Choose two examples that you can share, with which you can demonstrate the topic for the class.Reporter - Work with the other members of your group to create a presentation, using PowerPoint, which you will present to the class.
This interactive toolkit is designed to introduce concepts usually covered in a first course in linear algebra. The topics...
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This interactive toolkit is designed to introduce concepts usually covered in a first course in linear algebra. The topics are presented in an informal, casual manner. The student is periodically asked questions designed to stimulate him or her to review previous material, reflect over material just presented, or to consider alternative approaches. The feedback from the questions is immediate and usually offers extended explantion tailored to address reason why an correct or incorrect answer was selected. Since abstract concepts in mathematics are often best understood by forming an analogy with something concrete, LAVA makes use of interactive applets designed to foster a geometrical understanding of the concepts it presents.
This is a free, online wikibook, so its content is continually being updated and refined. According to the authors, "Linear...
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This is a free, online wikibook, so its content is continually being updated and refined. According to the authors, "Linear algebra is a branch of algebra in mathematics concerned with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are very important in modern mathematics. Linear algebra is widely used in abstract algebra and functional analysis. It has extensive applications in natural and social sciences, for both linear systems and linear models of nonlinear systems.״
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Written for undergraduate students and practicing engineers, this book provides an introduction to MATLAB and Simulink. Topics covered include programming in MATLAB, mathematic functions, MATLAB graphics, creating Simulink blocks and models, and simulation in Simulink.
MATLAB and Simulink are introduced and used to solve numerous examples throughout the book. In addition, the Symbolic Math Toobox is also briefly introduced and used in a chapter on symbolic mathematics.
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Summary: With a visual, graphical approach that emphasizes connections among concepts, this text helps students make the most of their study time. The authors show how different mathematical ideas are tied together through their zeros, solutions, and x-intercepts theme; side-by-side algebraic and graphical solutions; calculator screens; and examples and exercises. By continually reinforcing the connections among various mathematical concepts as well as different solution methods, the authors lead student...show mores to the ultimate goal of mastery and success in class. ...show less
The Law of Sines. The Law of Cosines. Complex Numbers: Trigonometric Form. Polar Coordinates and Graphs. Vectors and Applications. Vector Operations.
8. Systems and Matrices.
Systems of Equations in Two Variables. Systems of Equations in Three Variables. Matrices and Systems of Equations. Matrix Operations. Inverses of Matrices. Systems of Inequalities and Linear Programming. Partial Fractions.
9. Conic Sections.
The Parabola. The Circle and the Ellipse. The Hyperbola. Nonlinear Systems of Equations
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MATH110 - College Algebra
This course investigates the concepts of college algebra. The course covers the concepts of algebra, graphing and solution of linear and quadratic equations, inequalities and the solution of systems of linear equations. The course is organized into four distinct parts. The first part of the course covers the basic concepts involved in graphing points and linear equations. The second part of the course investigates the solution and graphing of inequalities and systems of linear equations. The third part of the course concentrates on the manipulation and use of exponential expressions and radicals. The final part of the course considers the solution of quadratic equations and their applications. Practical applications are provided throughout the course. There is careful attention to the presentation of concepts that will become important in the study of analytic geometry, trigonometry and calculus. The course assumes the student has completed MATH101 Introduction to College Algebra or an equivalent course and is completely comfortable with the language of algebra, equations and inequalities, polynomials, factoring, and rational expressions. If a lower-level math course has not been completed recently, we recommend that students take the 16-week session of MATH110. The eight-week session is recommended only for students with prior math experience and who have an adequate amount of time to pursue a highly-accelerated course of study in eight weeks.
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Transitioning to Proofs
One of the challenges in becoming a math major is learning the art of mathematical proof.
It is proof that distinguishes mathematics from other academic disciplines, giving our subject a permanence and transparency which makes mathematics truly timeless. Most students come to college having already exposed to a wide range of logical thinking processes, even if they haven't been aware of them. Being able to phrase and support logical arguments, however, is a skill that takes time and effort to master. As with any new skill, it will be your commitment to frequent and careful practice that will turn this new technique into second-nature.
As you transition into a proof-based class, your best resource is the professor of your class; this person can help you learn to read and write mathematical statements, and they can give you tips on how to approach certain standard proof practices. There are also a number of books which help students understand the basic structure of proofs. This PDF was a short guideline written by Erica Dohring ('14) which gives some ideas for how students might effectively transition into a proofs-based class; it also includes some suggestions for further reading as you explore this exciting new mathematical frontier
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a collaboration between a mathematician and a chemist, this text is geared toward advanced undergraduates and graduate students. Not only does it explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Rather than an exhaustive treatment, it represents an introduction that will appeal to a broad spectrum of students. Accordingly, the mathematics is kept as simple as possible.The first of the two-part treatment deals principally with the general properties of differintegral operators. The second half is mainly oriented toward the applications of these properties to mathematical and other problems. Topics include integer order, simple and complex functions, semiderivatives and semi-integrals, and transcendental functions. The text concludes with overviews of applications in the classical calculus and diffusion problems.
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dshaun88
The all-knowing solutions manual. Without it, millions of students across the globe wouldn't stand a chance. I remember a time - before my introduction to solutions manuals - when I actually did my homework. I read the assignment, researched the ques...
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College Algebra
9780077221287
ISBN:
0077221281
Edition: 3 Pub Date: 2008 Publisher: McGraw-Hill Companies, The
Summary: The Barnett Graphs & Modelsseries in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory. Many examples feature side-by-side algebraic and graphical solutions, and each is followed by a matched problem for the student to work. This active involvement in the learning process helps... students develop a more thorough understanding of concepts and processes. A hallmark of the Barnett series, the function concept serves as a unifying theme. A major objective of this book is to develop a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to proceed through this course with greater confidence and understanding as they first learn to recognize the graph of a function and then learn to analyze the graph and use it to solve the problem. Applications included throughout the text give the student substantial experience in solving and modeling real world problems in an effort to convince even the most skeptical student that mathematics is really useful.
Barnett, Raymond A. is the author of College Algebra, published 2008 under ISBN 9780077221287 and 0077221281. Three hundred ninety eight College Algebra textbooks are available for sale on ValoreBooks.com, one hundred twenty eight used from the cheapest price of $34.10, or buy new starting at $163.24 medited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
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and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform.
Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra. the intellectual equivalent of watching paint dry.
In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we've never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.
Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man's journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century's leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat's last theorem, that had seemed intractable before.
At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.
The international bestseller that inspired a major Nova special and sparked a new understanding of the universe, now with a new preface and epilogue.
Brian Greene, one of the world's leading string theorists, peels away layers of mystery to reveal a universe that consists of eleven dimensions, where the fabric of space tears and repairs itself, and all matter—from the smallest quarks to the most gargantuan supernovas—is generated by the vibrations of microscopically tiny loops of energy. The Elegant Universe makes some of the most sophisticated concepts ever contemplated accessible and thoroughly entertaining, bringing us closer than ever to understanding how the universe works.
There is an ill-concealed skeleton in the closet of physics: "As they are currently formulated, general relativity and quantum mechanics cannot both be right." Each is exceedingly accurate in its field: general relativity explains the behavior of the universe at large scales, while quantum mechanics describes the behavior of subatomic particles. Yet the theories collide horribly under extreme conditions such as black holes or times close to the big bang. Brian Greene, a specialist in quantum field theory, believes that the two pillars of physics can be reconciled in superstring theory, a theory of everything.
Superstring theory has been called "a part of 21st-century physics that fell by chance into the 20th century." In other words, it isn't all worked out yet. Despite the uncertainties--"string theorists work to find approximate solutions to approximate equations"--Greene gives a tour of string theory solid enough to satisfy the scientifically literate.
Though Ed Witten of the Institute for Advanced Study is in many ways the human hero of The Elegant Universe, it is not a human-side-of-physics story. Greene's focus throughout is the science, and he gives the nonspecialist at least an illusion of understanding--or the sense of knowing what it is that you don't know. And that is traditionally the first step on the road to knowledge. --Mary Ellen Curtin
Now in its third edition, Mathematical Concepts in the Physical Sciences provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.
Providing a comprehensive introduction to quantum field theory, this textbook covers the development of particle physics from its foundations to the discovery of the Higgs boson. Its combination of clear physical explanations, with direct connections to experimental data, and mathematical rigor make the subject accessible to students with a wide variety of backgrounds and interests. Assuming only an undergraduate-level understanding of quantum mechanics, the book steadily develops the Standard Model and state-of-the art calculation techniques. It includes multiple derivations of many important results, with modern methods such as effective field theory and the renormalization group playing a prominent role. Numerous worked examples and end-of-chapter problems enable students to reproduce classic results and to master quantum field theory as it is used today. Based on a course taught by the author over many years, this book is ideal for an introductory to advanced quantum field theory sequence or for independent study toolsAmazon Guest Review: Stephen Hawking Published in 1988, Stephen Hawking's A Brief History of Time became perhaps one of the unlikeliest bestsellers in history: a not-so-dumbed-down exploration of physics and the universe that occupied the London Sunday Times bestseller list for 237 weeks. Later successes include 1995's A Briefer History of Time, The Universe in a Nutshell, and God Created the Integers: The Mathematical Breakthroughs that Changed History. Stephen Hawking is Lucasian Professor of Mathematics at the University of Cambridge.
The third edition of this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics and many worked examples, it contains over 800 exercises. New stand-alone chapters give a systematic account of the 'special functions' of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, half of the exercises are provided with hints and answers and, in a separate manual available to both students and their teachers, complete worked solutions. The remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site,
Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive and comprehensible account of the physics of the universe. From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.
If Albert Einstein were alive, he would have a copy of The Road to Reality on his bookshelf. So would Isaac Newton. This may be the most complete mathematical explanation of the universe yet published, and Roger Penrose richly deserves the accolades he will receive for it. That said, let us be perfectly clear: this is not an easy book to read. The number of people in the world who can understand everything in it could probably take a taxi together to Penrose's next lecture. Still, math-friendly readers looking for a substantial and possibly even thrillingly difficult intellectual experience should pick up a copy (carefully--it's over a thousand pages long and weighs nearly 4 pounds) and start at the beginning, where Penrose sets out his purpose: to describe "the search for the underlying principles that govern the behavior of our universe." Beginning with the deceptively simple geometry of Pythagoras and the Greeks, Penrose guides readers through the fundamentals--the incontrovertible bricks that hold up the fanciful mathematical structures of later chapters. From such theoretical delights as complex-number calculus, Riemann surfaces, and Clifford bundles, the tour takes us quickly on to the nature of spacetime. The bulk of the book is then devoted to quantum physics, cosmological theories (including Penrose's favored ideas about string theory and universal inflation), and what we know about how the universe is held together. For physicists, mathematicians, and advanced students, The Road to Reality is an essential field guide to the universe. For enthusiastic amateurs, the book is a project to tackle a bit at a time, one with unimaginable intellectual rewards. --Therese Littleton
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Introduction to Mathematical Reasoning Numbers, Sets and Functions
9780521597180
ISBN:
0521597188
Pub Date: 1998 Publisher: Cambridge Univ Pr
Summary: Eccles, Peter J. is the author of Introduction to Mathematical Reasoning Numbers, Sets and Functions, published 1998 under ISBN 9780521597180 and 0521597188. Six hundred forty four Introduction to Mathematical Reasoning Numbers, Sets and Functions textbooks are available for sale on ValoreBooks.com, one hundred twenty one used from the cheapest price of $30.46, or buy new starting at $50
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In this tutorial on the inverse functions: y = 2 ^ x and y = log base 2 of x, I am using the program Camtasia Studio 2 to...
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In this tutorial on the inverse functions: y = 2 ^ x and y = log base 2 of x, I am using the program Camtasia Studio 2 to visually capture the key strokes of a cursor on a virtual TI 83 + calculator, and to also record the sound of my voice which is explaining the different steps taken. Students can regulate the flow of information in this tutorial as they wish. Furthermore, as suggested by Bill Hemme, my Program Director, I have included a page of Word, next to the calculator, which allows me to display the equations of the functions used, and some of the steps of computations written in Math Type. I would like to thank Nancy Doolittle for her technological assistance with this project.
Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java...
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Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law.
This section of a broader work, gives students a series of tutorial exercises in matrix multiplication. Topics include...
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This section of a broader work, gives students a series of tutorial exercises in matrix multiplication. Topics include matrix multiplication, vector multiplication, and the identity matrix. This page presents students with question that the enter answers to. Students are given feedback based on there entry.
This site includes more than 40 tutorials in Intermediate Algebra topics with practice tests and answer keys. The site is...
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This site includes more than 40 tutorials in Intermediate Algebra topics with practice tests and answer keys. The site is designed to assist the user in preparing for math placement tests and the math portion of the GRE.
This site contains tutorial lessons for College Algebra, Intermediate Algebra, Beginning Algebra, and Math for the Sciences....
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This site contains tutorial lessons for College Algebra, Intermediate Algebra, Beginning Algebra, and Math for the Sciences. Each lesson contains explanations, examples, and videos. There are also practice problems with complete solutions
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Based on Saxon's proven methods of incremental development and continual review strategies, the Saxon Calculus Homeschool Kit reviews key algebra, trigonometry and analytic geometry topics while introducing limits, functions, and the differentiation and integration of variables. This comprehensive text is ideal for future mathematicians, scientists and engineers!
Kit includes Student Textbook with 148 lessons, Homeschool Testing Book, Answer Key Booklet, and Solutions Manual with worked solutions to every problem in the textbook.
This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science
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My Math Lab: Student Access Kit (Hardcover)
This access kit will provide you with a code to get into MyMathLab, a personalized interactive learning environment, where you can learn mathematics and statistics at your own pace and measure your progress.In order to use My......more
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This access kit will provide you with a code to get into MyMathLab, a personalized interactive learning environment, where you can learn mathematics and statistics at your own pace and measure your progress.In order to use MyMathLab, you will need a CourseID provided by your instructor; MyMathLab is not a self-study product and does require you to be in an instructor-led course.MyMathLab includes:
Interactive
eBook
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This text from the author team of Aufmann and Nation offers the same engaging style and support for students as the Aufmann College Algebra series, all in a brief format that covers the entire course in a single semester. Interactive learning techniques incorporated throughout the text help students better understand concepts, focus their study habits, and achieve greater success.In this First Edition, the authors have also integrated many components into the textbook to help students diagnose and remediate weak algebra skills. Prerequisite review in the textbook and supporting materials allows students to fill in gaps in their mathematical knowledge, and keeps instructors from having to spend time on review. Extra support also comes from the Aufmann Interactive Method, featuring Try Exercises that allow students to practice math as it is presented and to more easily study for tests.
In the seven years since the publication of his first book, Functional Training for Sports, new understanding of functional anatomy created a shift in strength coaching. With this new material, Coach Boyle presents the continued evolution of functional training as seen by a leader in the strength and conditioning field. artists who need a quick and efficient way to join this brave new world will want 3D for Graphic Designers.
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Mathematics - Taught Programme
Overview
Programme Details
Mathematics is the cornerstone of modern society. From the security of passwords to weather predictions, from MRI scanners to CGI movies, mathematics underpins it all. Scientific advances, technological improvements, financial services, communications and engineering: all depend on sound mathematical foundations. The need for well trained mathematicians has never been greater.
Aims
The MSc in Mathematics prepares students for meeting this need. It aims to develop students' logical and analytical abilities, problem solving skills, and aptitude for thinking abstractly.
The programme gives students the flexibility to decide the topics of most interest to them. It is possible to study a wide range of subjects, or to focus more on a single subject area. A distinguishing feature from taught MSc's at other institutions is the ability to choose mathematical courses from a range of disciplines, including computing, physics and biology.
Contact Information
What you study
Syllabus
This is a 12 month programme, consisting of three courses in the first half session, three courses in the second half session, and a dissertation during the summer.
First Half Session – September to January
Two of the following three courses are required:
Analysis 1
Algebra 1
Geometry/Topology 1
One further course is required, to be chosen from the list above or from the following:
Measure Theory
Galois Theory
Nonlinear Dynamics and Chaos Theory
Logic and Categories
Modelling of Biological Systems
Reading Project 1
Second Half Session – February to June
Three courses from the following must be chosen:
Analysis 2
Algebra 2
Geometry/Topology 2
Knot Theory
Geometry
Number Theory
Algebraic Topology
Reflection Groups
Mathematical Modelling
Reading Project 2
Summer Project/Dissertation - June to September
The summer project is taken under the supervision of an assigned academic supervisor in the department. Students will investigate in depth a specialist topic, write a dissertation on this topic which is mathematically rigorous and of a high standard, and present the topic to a board of examiners.
Assessment
Assessment is by course work, by written examination or by a combination of these as prescribed for each course. The summer project will be assessed by dissertation. The degree of MSc shall not be awarded to a candidate who fails to complete the summer project at an appropriate standard, irrespective of their performance in other courses.
Careers
Careers
"I have worked for many years in Investment Banking including twelve years as a CFO in Asia and Europe. In my work I continue to appreciate the good foundation my mathematical studies have given me. This is both in terms of a disciplined approach to problem solving as well as dealing with today's complex banking products. It has also been essential that strong mathematical knowledge extends across my teams as they perform critical risk management functions.
I would encourage anyone with an interest in a career in Finance to study mathematics as a route for developing the skills and understanding required. Furthermore, in-depth mathematical knowledge is an ideal introduction to a wide range of banking careers from accounting and control to trading and structuring derivatives or other investment products."
Richard Croydon Private Investment COO
Former CFO Global Equities, ABN AMRO Bank N.V. Former CFO Asia, ING
What you need
Requirements
The minimum requirement is the equivalent of an upper second class UK Honours degree in Mathematics.
It is important when submitting an application that you ensure you have completed all the necessary sections and enclosed all the relevant documentation to ensure that your application can be processed as quickly as possible.
Even if you have been educated in the medium of English you must meet our English Language requirements. These are located at This programme requires that you meet the 'Postgraduate Standard' level of English proficiency. If you are in doubt about your proficiency in English, contact the British Council office or its equivalent in your country. If your first language is not English, it is important that your proficiency in English is good in order for you to study successfully at the University of Aberdeen . Without this ability you will find great difficulty in understanding lectures, producing written work and sitting examinations.
We have one intake of students each year in September. Late applications may be asked to wait until the next intake should the programme coordinator feel there is insufficient time to consider the application. Prospective students who require a visa to study in the UK are advised to apply as early in the year as possible to secure a place. Applications received after 30th June from students who need to apply for a visa to study in the UK will not be processed for entry in September of that year but may be considered for entry the following year as appropriate.
It is important to note that the programmes of postgraduate study at the University of Aberdeen are very competitive and the entry requirements stated are a guide to the minimum requirements, but do not guarantee entry.
Fees and Funding
Fees
Information on tuition fees, including the current fee level, can be found on the University Registry website.
Funding
Funding opportunities can be found in our searchable Funding Database. You are advised to search the database as a broad range of funding exists much of which you may be eligible for.
University of Aberdeen Alumni Discount Scheme
The University of Aberdeen is very pleased to offer a 20% discount on postgraduate tuition fees for all alumni who have graduated (or about to graduate) with a degree from the University of Aberdeen. More Information
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Making the transition to calculus means being prepared to grasp bigger and more complex mathematical concepts. Precalculus: Functions and Graphs is designed to make this transition seamless, by focusing now on all the skills that you will need in the future. The foundation for success begins with preparation and Precalculus: Functions and Graphs will help you succeed in this course and beyond.
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Written by top researchers, this text blends theory and practice. It covers the modern topics of parallel algorithms, concurrency and recurrency. A McGraw-Hill/MIT Press collaboration, the text is designed for both the instructor and the student. It offers a flexible organization with self-contained chapters, and it provides an introduction to the necessary mathematical analysis. Introduction to Algorithms contains sections that gently introduce mathematical techniques for students who may need help. This material takes students at an elementary level of mathematical sophistication and raises them to a level allowing them to solve algorithmic problems. Simple, easy-to-do exercises, as well as more thoughtful, step-by-step case-generated problems are included. The book features standard analytic notation and includes trimmed-down, easy-to-read pseudocode.
Review:
If you had to buy just one text on algorithms, Introduction to Algorithms is a magnificent choice. The book begins by considering the mathematical foundations of the analysis of algorithms and maintains this mathematical rigor throughout the work. The tools developed in these opening sections are then applied to sorting, data structures, graphs, and a variety of selected algorithms including computational geometry, string algorithms, parallel models of computation, fast Fourier transforms (FFTs), and more.
This book's strength lies in its encyclopedic range, clear exposition, and powerful analysis. Pseudo-code explanation of the algorithms coupled with proof of their accuracy makes this book is a great resource on the basic tools used to analyze the performance of algorithms
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Course Description
Description
This proof-based geometry course, based on a popular classic textbook, covers concepts typically offered in a full-year honors geometry course. To supplement the lessons in the text book, videos, online interactives, assessments and projects provide students an opportunity to develop mathematical reasoning, critical thinking skills, and problem solving techniques to investigate and explore geometry. Additionally, students are introduced to a dynamic software tool, GeoGebra, through projects that they create.
This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.
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Elementary and Inter. Algebra : Graphs and Models - 4th edition
Summary: TheBittinger Graphs and Models Serieshelps readers learn algebra by making connections between mathematical concepts and their real-world applications. Abundant applications, many of which use real data, offer students a context for learning the math. The authors use a variety of tools and techniques-including graphing calculators, multiple approaches to problem solving, and interactive features-to engage and motivate all types of learners2634033.39 +$3.99 s/h
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From the Student Library, click on "Online Math Coaching" and this screen will appear
Note the Live Classroom days/hours available for your course as well as the 800 call in number and code
Note the Live Tutorial hours available for your course
What to Expect from a Math Coach
Math Coaches will write on the whiteboard as they answer your question
Math Coaches can use your problem or a similar one to teach you
Math Coaches will ask questions to guide you into understanding and to check that you are understanding
You will be doing most of the work
Center for Mathematics Excellence
Live Classroom: One Math Coach for Many Students
Join Live Class, where multiple students can participate in a coaching session
Live teleconference available with fellow students and math coach
Toll-free dial-in number
Only available to students who are currently enrolled in a math course
During class hours, just click Join Live Class and call the 800 number
Center for Mathematics Excellence
Join Live Tutorial: One-to-One Math Coaching Choose "Join Live Tutorial" Choose from the list of active math coaches (hint: choose the room with the smallest queue!) You may see a welcome screen, or you may see the math coach working with a student. Press the "Ask a question" button to begin. Note: Join Live Tutorial is only available for Algebra & Statistic students Center for Mathematics Excellence
Working with a Math Coach Type your question and click "Submit Your Question." You can use the toolbar on the left side of the screen to add fractions or other math symbols. This puts your question in the queue. Math Coaches answer questions in the order they are submitted. When the math coach is ready to work with you, a bell will ring, and you can both write on the screen together. Center for Mathematics Excellence 3 2 1
Tips " Float" button detaches the screen from the background to enlarge it. After tutorial is finished, you can continue to view or print out the screen. To return to the queue, click "View Tutorial." Center for Mathematics Excellence
Support
Student Tech Support is available 24/7: (877) 832-4867
Whiteboard Tech Support - click located on the CME homepage. After you fill out the form, LSI will contact you to resolve your issue.
For questions or comments about our services, please use the CME email address: [email_address]
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Web Resources
Lesson Plans
Title: Pedal Power
Description:
In this lesson, students investigate slope as a rate of change. Students compare, contrast, and make conjectures based on distance-time graphs for three bicyclists climbing to the top of a mountain 14: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of linear function in terms of the situation it models and in terms of its graph or a table of values. [8-F4] [MA2013] (8) 15: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. [8-F5] [MA2013] AL1 (9-12) 46: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [S-ID7]
Pedal Power
In this lesson, students investigate slope as a rate of change. Students compare, contrast, and make conjectures based on distance-time graphs for three bicyclists climbing to the top of a mountain.
Podcasts
Title: Math in Video Games
Description:
The teams use algebra to save their spaceship in the Asteroids game 27: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. [8-SP3] [MA2013] AL1 (9-12) 46: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. [S-ID7] [MA2013] GEO (9-12) 31: Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). [G-GPE5]
Title: Math in Special Effects
Description:
Jeremy Chernick from J&M Special Effects discusses how he ended up creating effects as a career, then introduces a challenge about the algebra behind lighting high-speed effects like explosions.
Standard(s):8) 7: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5]
Math in Special Effects
Jeremy Chernick from J&M Special Effects discusses how he ended up creating effects as a career, then introduces a challenge about the algebra behind lighting high-speed effects like explosions.
Teacher Tools
Title: Algebra.Help
Description:
Lesson plans,calculators, worksheets and everything else to help with teaching algebra.
Standard(s): [MA2013] (8) 1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. [8-NS1] [MA2013] (8) 2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). [8-NS2] [MA2013] (8) 7: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5] [MA2013] (8) 9: Solve linear equations in one variable. [8-EE7] [MA2013] AL1 (9-12) 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. [N-RN1]
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A History of Mathematics: An Introduction
This text grew from the authors' conviction that both prospective school teachers and college teachers of maths need a background in history to be ...Show synopsisThis text grew from the authors' conviction that both prospective school teachers and college teachers of maths need a background in history to be more effective as instructors in the classroom. Prospective instructors gain an appreciation of the contributions of all cultures, and this text explains how mathematics developed over the centuries. Also suitable for those studying maths and science at degree level1387004Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780321387004
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The High School Math Program is designed to maximize the acquisition of both skills and concepts that the students must have upon finishing their high school. Four dimensions of understanding are emphasized in the high school mathematics: (1) skill in carrying out various algorithms; (2) developing and using mathematics properties and relationships; (3) applying mathematics in realistic situations; and (4) representing or picturing mathematical concepts.
The high school mathematics thoroughly integrates and makes connections to other areas of mathematics (vertically articulated). Furthermore, it makes connections to other disciplines (e.g. language, science) and to the real world. The following are the Math courses offered in the high school: First year – Algebra; Second year – Geometry; Third year – Advanced Algebra; and Fourth year – Functions, Statistics and Trigonometry.
By Year Level Description
The first year mathematics is Algebra. The students are introduced to the language of Algebra and its uses.
The second year mathematics is Geometry. The students are introduced to making conjectures and proofs; applying definitions, postulates, and theorems in proving statements; draw, construct, and transform geometric figures.
The third year mathematics is Advanced Algebra. The students are engaged to a deeper understanding and application of concepts in Algebra.
The fourth year mathematics is Functions, Statistics, and Trigonometry. The students are lead to integrate statistical, algebraic, and trigonometric concepts through its work with functions. They will also be introduced to combinatorics and probability.
Honor's Classes
The Honor's Classes will have additional competencies. They are as follows: Analytic Geometry (Second Year), Trigonometry (Third Year), and Calculus (Fourth Year).
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Geometric Algebra for Computer Science (Revised Edition), (The Morgan Kaufmann Series in Computer Graphics) for an Amazon Gift Card of up to £2.30, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book DescriptionMore About the Author
Product Description
Review
Within the last decade, Geometric Algebra (GA) has emerged as a powerful alternative to classical matrix algebra as a comprehensive conceptual language and computational system for computer science. This book will serve as a standard introduction and reference to the subject for students and experts alike. As a textbook, it provides a thorough grounding in the fundamentals of GA, with many illustrations, exercises and applications. Experts will delight in the refreshing perspective GA gives to every topic, large and small. -David Hestenes, Distinguished research Professor, Department of Physics, Arizona State University Geometric Algebra is becoming increasingly important in computer science. This book is a comprehensive introduction to Geometric Algebra with detailed descriptions of important applications. While requiring serious study, it has deep and powerful insights into GA's usage. It has excellent discussions of how to actually implement GA on the computer. -Dr. Alyn Rockwood, CTO, FreeDesign, Inc. Longmont, Colorado
About the Author
Daniel Fontijne holds a Master's degree in artificial Intelligence and a Ph.D. in Computer Science, both from the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision.
Geometric Algebra (GA) is a unifying mathematical language that should be taught instead of or at least in combination with traditional vector analysis. Most other books on GA are aimed at Physicists. This book is a better match for Engineers and Programmers. The authors are all active researchers in applications of GA. They have done a comprehensive and up to date job of collecting, organizing and presenting the material for both beginners and those who follow the development of GA on the web. The examples and problems use GAViewer, an easy to learn programming language with an Open GL view window that can be downloaded for free from the book website. Using GAViewer with the book is very good way to learn GA, especially the 5D Conformal model of 3D space. The authors hold nothing back. Between the book, the code and the website everything is there to make learning GA fun and useful. I highly recommend this book.
12 of 12 people found the following review helpful
5.0 out of 5 starsAn excellent introduction to the subject.5 Sep 2009
By Peeter Joot - Published on Amazon.com
Format:Hardcover
The book Geometric Algebra For Computer Science, by Dorst, Fontijne, and Mann has one of the best introductions to the subject that I have seen.
It contains particularly good introductions to the dot and wedge products and how they can be applied and what they can be used to model. After one gets comfortable with these ideas they introduce the subject axiomatically. Much of the pre-axiomatic introductory material is based on the use of the scalar product, defined as a determinant. You'll have to be patient to see where and why that comes from, but this choice allows the authors to defer some of the mathematical learning overhead until one is ready for the ideas a bit better.
Having started study of the subject with papers of Hestenes, Cambridge, and Baylis papers, I found the alternate notation for the generalized dot product (L and backwards L for contraction) distracting at first but adjusting to it does not end up being that hard.
This book has three sections, the first covering the basics, the second covering the conformal applications for graphics, and the last covering implementation. As one reads geometric algebra books it is natural to wonder about this, and the pros, cons and efficiencies of various implementation techniques are discussed.
There are other web resources available associated with this book that are quite good. The best of these is GAViewer, a graphical geometric calculator that was the product of some of the research that generated this book. Performing the GAViewer tutorial exercises is a great way to build some intuition to go along with the math, putting the geometric back in the algebra.
There are specific GAViewer exercises that you can do independent of the book, and there is also an excellent interactive tutorial available. Browse the book website, or Search for '2003 Game Developer Lecture, Interactive GA tutorial. UvA GA Website: Tutorials'. Even if one decided not to learn GA, using this to play with the graphical cross product manipulation, with the ability to rotate viewpoints, is quite neat and worthwhile.
3 of 4 people found the following review helpful
5.0 out of 5 starsvery good text10 May 2010
By T. Czyczko - Published on Amazon.com
Format:Hardcover
This is the text I would first recommend to anyone involved in geometrical programming who would like to learn geometrical algebra.
4 of 13 people found the following review helpful
3.0 out of 5 starsok, but...5 Oct 2010
By Fuga Federico - Published on Amazon.com
Format:Hardcover|Verified Purchase
It's a good book, but the mathematics is poorly treated, not enough rigorous as would be expected.
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Beginning Pre-Calculus for Game Developers
9781598632910
ISBN:
1598632914
Edition: 1 Pub Date: 2006 Publisher: Course Technology
Summary: Beginning Pre-Calculus for Game Developers provides entertaining, hands-on explanations of topics central to calculus as related to game development. It explains the mathematics and programming involved in developing nine computer programming applications furnished with the book's CD-ROM. Begin by working your way through first semester calculus topics and then use your new math skills to create programs that apply e...ach topic. Beginning Pre-Calculus presents math topics in a method that is direct, easy-to-understand, and pertinent to all studies related to calculus math.
Flynt, John P. is the author of Beginning Pre-Calculus for Game Developers, published 2006 under ISBN 9781598632910 and 1598632914. Two hundred forty six Beginning Pre-Calculus for Game Developers textbooks are available for sale on ValoreBooks.com, one hundred five used from the cheapest price of $0.01, or buy new starting at $20.77
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Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton's engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.
Elementary Number Theory, 7e, by David M. Burton
Table of Contents
Preface
New to this Edition
1 Preliminaries
1.1 Mathematical Induction
1.2 The Binomial Theorem
2 Divisibility Theory in the Integers
2.1 Early Number Theory
2.2 The Division Algorithm
2.3 The Greatest Common Divisor
2.4 The Euclidean Algorithm
2.5 The Diophantine Equation
3 Primes and Their Distribution
3.1 The Fundamental Theorem of Arithmetic
3.2 The Sieve of Eratosthenes
3.3 The Goldbach Conjecture
4 The Theory of Congruences
4.1 Carl Friedrich Gauss
4.2 Basic Properties of Congruence
4.3 Binary and Decimal Representations of Integers
4.4 Linear Congruences and the Chinese Remainder Theorem
5 Fermat's Theorem
5.1 Pierre de Fermat
5.2 Fermat's Little Theorem and Pseudoprimes
5.3 Wilson's Theorem
5.4 The Fermat-Kraitchik Factorization Method
6 Number-Theoretic Functions
6.1 The Sum and Number of Divisors
6.2 The Möbius Inversion Formula
6.3 The Greatest Integer Function
6.4 An Application to the Calendar
7 Euler's Generalization of Fermat's Theorem
7.1 Leonhard Euler
7.2 Euler's Phi-Function
7.3 Euler's Theorem
7.4 Some Properties of the Phi-Function
8 Primitive Roots and Indices
8.1 The Order of an Integer Modulo n
8.2 Primitive Roots for Primes
8.3 Composite Numbers Having Primitive Roots
8.4 The Theory of Indices
9 The Quadratic Reciprocity Law
9.1 Euler's Criterion
9.2 The Legendre Symbol and Its Properties
9.3 Quadratic Reciprocity
9.4 Quadratic Congruences with Composite Moduli
10 Introduction to Cryptography
10.1 From Caesar Cipher to Public Key Cryptography
10.2 The Knapsack Cryptosystem
10.3 An Application of Primitive Roots to Cryptography
11 Numbers of Special Form
11.1 Marin Mersenne
11.2 Perfect Numbers
11.3 Mersenne Primes and Amicable Numbers
11.4 Fermat Numbers
12 Certain Nonlinear Diophantine Equations
12.1 The Equation
12.2 Fermat's Last Theorem
13 Representation of Integers as Sums of Squares
13.1 Joseph Louis Lagrange
13.2 Sums of Two Squares
13.3 Sums of More Than Two Squares
14 Fibonacci Numbers
14.1 Fibonacci
14.2 The Fibonacci Sequence
14.3 Certain Identities Involving Fibonacci Numbers
15 Continued Fractions
15.1 Srinivasa Ramanujan
15.2 Finite Continued Fractions
15.3 Infinite Continued Fractions
15.4 Farey Fractions
15.5 Pell's Equation
16 Some Recent Developments
16.1 Hardy, Dickson, and Erdös
16.2 Primality Testing and Factorization
16.3 An Application to Factoring: Remote Coin Flipping
16.4 The Prime Number Theorem and Zeta Function
Miscellaneous Problems
Appendixes
General References
Suggested Further Reading
Tables
Answers to Selected Problems
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Intermediate Algebra (Cloth) - 2nd edition
ISBN13:978-0073312682 ISBN10: 0073312681 This edition has also been released as: ISBN13: 978-0073028729 ISBN10: 007302872X
Summary: Miller/O'Neill/Hyde, built by teachers just like you, continues continues to offer an enlightened approach grounded in the fundamentals of classroom experience in the 2nd edition of Intermediate Algebra. The practice of many instructors in the classroom is to present examples and have their students solve similar problems. This is realized through the Skill Practice Exercises that directly follow the examples in the textbook. Throughout the text, the authors have integr...show moreated many Study Tips and Avoiding Mistakes hints, which are reflective of the comments and instruction presented to students in the classroom. In this way, the text communicates to students, the very points their instructors are likely to make during lecture, helping to reinforce the concepts and provide instruction that leads students to mastery and success. The authors included in this edition, Problem-Recognition exercises, that many instructors will likely identify to be similar to worksheets they have personally developed for distribution to students. The intent of the Problem-Recognition exercises, is to help students overcome what is sometimes a natural inclination toward applying problem-sovling algorithms that may not always be appropriate. In addition, the exercise sets have been revised to include even more core exercises than were present in the first edition. This permits instructors to choose from a wealth of problems, allowing ample opportunity for students to practice what they learn in lecture to hone their skills and develop the knowledge they need to make a successful transition into College Algebra. In this way, the book perfectly complements any learning platform, whether traditional lecture or distance-learning; its instruction is so reflective of what comes from lecture, that students will feel as comfortable outside of class, as they do inside class with their instructor. For even more support, students have access to a wealth of supplements, including McGraw-Hill's online homework management system, MathZone70 +$3.99 s/h
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2006-12-30 Hardcover Good Names on inside cover and numbers on bookedge; no other internal marking/highlighting.
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MYSMILLINGBOOK MIAMI GARDENS, FL
2006 Hard cover 2nd ed. Very good in very good dust jacket. Glued binding. Paper over boards. 842 p. Contains: Illustrations. Audience: General/trade. BOX# ALI051011
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Calculus Limits: A Graphical Approach This video explains the graphical approach to determine a limit. There are two ways to determine a limit: a numerical approach or a graphical approach. In the graphical approach, we analyze the graph of the function to determine the points that each of the one-sided limits approach. (3:02)Percent and Decimals In this video the view is walked through the process of expressing percentages as decimals and expressing decimals as percentages with examples with the Paint Program.Mobl21 - Introduction 21st Century e-Teacher Chris Jones introduces the power of mobile learning by using the Mobl21 authoring and distribution application. Run time 01:03. Author(s): No creator set
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Mobl21 - Groups and Users 21st Century e-Teacher Chris Jones explains how to set-up Groups and Users within the Mobl21 authoring and distribution application. Run time 04:18. Author(s): No creator set
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Mobl21 - Behind the Scenes of the Mini-module Take a peek at the making of the videos for the Mobl21 Mini-module by our intrepid 21st Century e-Teacher Chris Jones. You'll be amazed at how a home set-up resulted in his great work. Run time 01:00. Author(s): No creator set
Plagiarism--Before He Cheats This well-produced video showcases the song "Before He Cheats," and is a parody about plagiarism and cheating in school. This is appropriate for students in grades 7-12 (USA). 3:25 min. Author(s): No creator set
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Laurie Anderson's Dirtday! Performance artist Laurie Anderson talks about her new work Dirtday! She begins her US tour at Cal Performances, located on the campus of UC Berkeley, Tuesday, September 18, 2012. Author(s): No creator set
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Redwoods at Redwood Forest Take a walk with host Jim Welch and naturalist Rudy Mancke in this video segment from NatureScene as they describe the resilience and
longevity of redwood trees. Learn about requirements for redwood growth, methods of reproduction, and the effects of natural fires. Closed captioning included. Run time 04:02.
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What is a Noun? This video defines a noun. People, places, events, idea, and things are also defined and then given a picture example. Singular, plural, irregular, common and proper nouns are defined as well. Some of these examples are not developmentally appropriate for younger students (5:24). Author(s): No creator set
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The Washer Women This is a traditional Irish folk song performed by Andre Rieu on the violin. The video starts as solo violin then the orchestra joins in and the tempo starts to fly! The video could also be used with younger students while teaching rhythm or an Irish Dance. (2:16) Author(s): No creator set
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Introduction to Catholic Moral Theology Notre Author(s): admin
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Copyright 2012, by the Contributing AuthorsEpidemiology: An introduction Public health interventions need to be built on an evidence base and part of this evidence comes from epidemiology: the study of how and why diseases occur. Epidemiology is a bit like a game of detection. It involves identifying diseases, finding out which groups of people are at risk, tracking down causes and so on. This unit looks at some key types of data used in epidemiology, such as statistics on death and ill health, and introduces some techniques used in analysing data. Author(s): The Open University
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These applets provide interactive student activities to make connections between the graphical and analytical interpretation of "completing the square" and writing the equation of a parabola in general and standard form.
Gauss-Kronrod Integration is an adaptation of Gaussian quadrature used on some graphing calculators. This Java applet outlines the mathematical computations involved and visually demonstrates the process the calculator uses to evaluate the integral.
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Course Description
MAED 533 Developing Geometrical Thinking K-12
2 cr.
Survey of concepts, principles and mathematical processes in the geometry curriculum; common difficulties encountered by students and alternatives for structuring learning; and applications of technology, materials and resources for teaching geometry.
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A Mathematical MATHEMATICAL VIEW OF OUR WORLD helps students understand and harness the power of mathematics in their present studies and future careers. Designed for a liberal arts mathematics course that has a strong focus on applications, this student-friendly textbook helps students see the beauty and power of mathematics as it is applied to the world around them. Recognizing that quantitative literacy is essential in an increasing number of professional fields as well as in the daily decision-making and communication of informed citizens, the book develops students' mathematical self-confidence and prepares them to use mathematics in the future. In writing the book, the authors endeavored to be faithful to the recommendations of such professional mathematics associations as the MAA, AMATYC, and NCTM. Overall, students will recognize the connections, patterns, and significance of the mathematics they study, and see that mathematics has a meaningful place in their lives.
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This lucid and balanced introduction for first year engineers and applied mathematicians conveys the clear understanding of the fundamentals and applications of calculus, as a prelude to studying more advanced functions. Short and fundamental diagnostic exercises at the end of each chapter test comprehension before moving to new material. Provides... more....... more...
Conventional calculus is too hard and too complex. Students are forced to learn too many theorems and proofs. In Free Calculus, the author suggests a direct approach to the two fundamental concepts of calculus — differentiation and integration — using two inequalities. Regular calculus is condensed into a single concise chapter. This... more...
This book is a detailed study of Gottfried Wilhelm Leibniz's creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known "calculi" Leibniz created such as the Analysis Situs,...Mathematics and engineering are inevitably interrelated, and this interaction will steadily increase as the use of mathematical modelling grows. Although mathematicians and engineers often misunderstand one another, their basic approach is quite similar, as is the historical development of their respective disciplines. The purpose of this Math Primer... more...
Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting... more...
This collection of peer-reviewed conference papers provides comprehensive coverage of cutting-edge research in topological approaches to data analysis and visualization. It encompasses the full range of new algorithms and insights, including fast homology computation, comparative analysis of simplification techniques, and key applications in materials... more...
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More About
This Textbook
Overview
Dugopolski's Trigonometry, Third Edition givesopolski's emphasis on problem solving and critical thinking, which is enhanced by the addition of hundreds of exercises in this edition.
Related Subjects
Meet the Author
2. Graphs of the Trigonometric Functions.
Graphs of the Sine and Cosine Functions.
The General Sine Wave.
Graphs of the Tangent and Cotangent Functions.
Graphs of the Secant and Cosecant Functions.
Combining Functions 10, 2007
A reviewer
I took Trigonometry in high school, but when Caculus II came around, I completely forgot my trig identities. So I went on and purchased this book. It's a great mathematical textbook! It can be used by the math whiz kids, and math phobes too! Great compliment to any library.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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More About
This Textbook
Overview
This book deals with the twistor treatment of certain linear and non-linear partial differential equations in mathematical physics. The description in terms of twistors involves algebraic and differential geometry, and several complex variables, and results in a different kind of setting that gives a new perspective on the properties of space-time and field theories. The book is designed to be used by mathematicians and physicists and so the authors have made it reasonably self-contained. The first part contains a development of the necessary mathematical background. In the second part, Yang-Mills fields and gravitational fields (the basic fields of contemporary physics) are described at the classical level. In the final part, the mathematics and physics are married to solve a number of field-theoretical problems.
Editorial Reviews
From the Publisher
"... skillfully written. It will serve as a relatively accessible introduction to twistor theory for many readers who have not studied the subject before. Others will find it useful as a refresher and as a source of many valuable insights." Nature
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Limits, Limits Everywhere: The Tools of Mathematical Analysis for an Amazon Gift Card of up to £7.00, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description{"currencyCode":"GBP","itemData":[{"priceBreaksMAP":null,"buyingPrice":19.26,"ASIN":"0199640084","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":11.92,"ASIN":"0199661324","isPreorder":0}],"shippingId":"0199640084::SJRKNX4jzP9REkaEr08QAaaxDEvPs4zQhhvZPHsZOu36eKqsorCDlS6LejLLaiLHkrPsBJ7OpTyAYtKpxjcpW%2FljaU85bAiT,0199661324::fmVNUcXxqrp%2FBekDs9h7fNeRyefjhiwj0wCjImB%2FIl4fW8IOCJnpg19CIURYj2Jrn3xxSPIY4IAjhj1z7CdZT37r3%2BXVlxf does not offer an easy ride but its informal and enthusiastic literary style hold ones attention. Perhaps mindful of the content of much current popular mathematical exposition, the author draws many illustrations from number theory. (Geoffrey Burton, LMS Newsletter)
The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow. (Luis Sanchez-Gonzalez, the European Mathematical Society)
Written in a style that is easy to read and follow, the author gives clear and succinct explanations and meets his desire for this to be between a textbook and a popular book on mathematics. (John Sykes, Mathematics in Schools)
This is an excellent book which should appeal to teachers and pre-University or undergraduate students looking for a hands-on introduction to mathematical analysis. (Mario Cortina Borja, Significance)
The book is devoted to the discussion of one of the most difficult concepts of mathematical analysis, the concept of limits. The presentation is instructive and informal. It allows the author to go much deeper than is usually possible in a standard course of calculus. Moreover, each portion of the material is supplied by an explanation why and what for it is necessary to study (and to teach) the corresponding part of calculus ... the book can be recommended for interested students as well as for teachers in mathematics. (Zentralblatt MATH)
About the Author
David Applebaum obtained his PhD at the University of Nottingham in 1984. After postdoctoral appointments in Rome and Nottingham, he became a lecturer in mathematics at Nottingham Trent University (then Trent Polytechnic) in 1987 and was promoted to reader in 1994 and to a chair in 1998. He was Head of Department 1998-2001. He left Nottingham Trent for a chair in Sheffield in 2004 and served as Head of Department of Probability and Statistics there from 2007-10.
| 677.169 | 1 |
College Geometry A Discovery Approach With the Geometer's Sketchpad
9780321046246
ISBN:
0321046242
Edition: 2 Pub Date: 2000 Publisher: Addison-Wesley
Summary: College Geometry is Moments for Discovery, that use drawing, computational, o...r reasoning experiments to guide students to an often surprising conclusion related to section concepts; and (2) More than 650 problems were carefully designed to maintain student interest.
Kay, David C. is the author of College Geometry A Discovery Approach With the Geometer's Sketchpad, published 2000 under ISBN 9780321046246 and 0321046242. Four hundred eighty nine College Geometry A Discovery Approach With the Geometer's Sketchpad textbooks are available for sale on ValoreBooks.com, one hundred thirty used from the cheapest price of $71.59, or buy new starting at $90
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Pacemaker Geometry
Help your students grasp geometric concepts Through a clear and thorough presentation, this program fosters learning and success for students of all ...Show synopsisHelp your students grasp geometric concepts Through a clear and thorough presentation, this program fosters learning and success for students of all ability levels with extensive skills practice, real-life connections, projects, and study aids. The accessible format helps students gain the understanding and confidence they need to improve their performance on standardized tests. Margin notes provide links to postulates and concepts previously taught; theorem boxes help students identify the big ideas in geometry. Featured lessons address calculator usage, applications, as well as paragraph proofs and constructions. Pre-taught vocabulary provides students with relevant background. Lexile Level 670 Reading Level 3-4 Interest Level 6-1230238376 This book shows signs of shelf wear and may...Good. 0130238376 This book shows signs of shelf wear and may contain a school stamp, sticker, class set number, or other markings on the inside or outside cover. This book may also contain highlighting, writing on the inside pages, and other markings. Fast shipping! ! !
Description:Good. Former school library book. has name of school printed in...Good. Former school library book. has name of school printed in front of book pages twice and also a big stamp of THIS BOOK IS PROPERTY OF with lines underneath. Otherwise no known markings/highlightings in books. Tulsa's best used bookstore. Located on South Mingo Road since 1991. No-hassle return policy if not completely satisfied.
Description:Fair. 0130238376 WE HAVE NUMEROUS COPIES-HARDCOVER, some pages...Fair. 0130238376 WE HAVE NUMEROUS COPIES-HARDCOVER, some pages have mild rippling from liquid exposure but does not affect usability, otherwise book is in good condition with moderate wear to cover/edges/corners-has cardboard showing on corners.
Description:Very Good. 0130238414 MULTIPLE COPIES AVAILABLE-Very Good...Very Good. 01302384 0130238414 No excessive markings and minimal...Very Good. 0130238414
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Foundations of Geometry
This text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis ...Show synopsisThis text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis and abstract algebra). It fully implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers. Foundations of Geometry particularly teaches good proof-writing skills, emphasises the historical development of geometry, and addresses certain issues concerning the place of geometry in human culture.Hide synopsis 9780321762245....New in new dust jacket. Brand New as listed. ISBN 9780321762245. Clean! Out of sight Shipping & Customer Service! We process all orders same day! !
Description:New. 0136020585 BRAND NEW. PLASTIC WRAPPED. We are a tested and...New. 0136020585 Foundations of Geometry
I am a high school math teacher and bought this book to help me with a course for my master's degree. This book came recommended by others. It is easy to read and understand! I believe that even a person that does not have a math background would be able to pick up on the main ideas that are being presented. The differences between Euclidean and non-Euclidean studies of geometry were clearly explained. A
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Advanced math courses are nothing like what you're doing now. If you like grinding through step after step of algebra, you'll miss that. They're extremely abstract. Read the wiki page on topology and as many related/linked articles as possible. You'll quickly get an idea of what I mean by abstract.
If you want something more practical but still math-centric, try electrical engineering.
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My print column this week explores a new search engine, Wolfram Alpha, that aims to make the world's information computable. It could also make the world's math problem sets and tests computable, by solving tough problems, including in calculus, and showing its work.
Some teachers see Wolfram Alpha as a tool liberating them and their students to focus on broader concepts, just as calculators obviated slide-rule instruction. It could also push math into a more visual realm and away from abstract notation, thanks to its plethora of graphs and charts. "Graphical aspect: I'm wondering how much mathematical notation will survive this big push of graphing and animation," said Rich Beveridge, a math instructor at Clatsop Community College who has blogged about the new site.
However, Roger Howe, a math professor at Yale university, worries that the basics will be forgotten. "Mathematics doesn't really become real unless students have a fairly direct contact with it," Howe says. "Doing a reasonable amount of computation seems to be important for mastering mathematics."
It's unclear how much will be lost, and how much that matters. "One worries we'll lose the underlying intuition," said Donald Berry, chairman of the department of biostatistics at the University of Texas M.D. Anderson Cancer Center. "I worry about that, but I'm not sure how important that is. We as a species have come to a point where we can do things because of what others have built for us."
Maria Andersen, a mathematics instructor at Muskegon Community College whose blog has hosted a lively discussion on Wolfram Alpha, is excited to use the new search engine in instruction. "I do see it as being a fantastic tool we can use to explore concepts," Andersen said. "I can't morally imagine walking into a classroom and having a student say, 'Why do I need to take a cube of a binomial when Wolfram Alpha can do it for me?' "
She added that teachers would have to change their homework. "I would say that at least 50% of the standard homework problems and assessment questions that would be assigned with traditional textbooks could be answered by Wolfram Alpha," Andersen said. "… find a math book, open it to a section of problems, and you'll quickly find lots of examples that Wolfram Alpha will do."
But Andersen worries about becoming reliant on a single online tool: "What if Wolfram Alpha disappears, after we all shifted to use it?"
"It's very flattering if people care enough about one's tools to be concerned about their longevity," says Stephen Wolfram, the founder and chief executive of Wolfram Research. "What can one guarantee in this world?"
He sees his new tool as a way to broaden access to math and science. "The more people have ready access to knowledge and the more they have the power to do things like the experts do, the more they can feel empowered and get motivated to understand what's going on," he said.
Some calculation systems can work too much like a black box, though, according to Colm Mulcahy, a mathematician at Spelman College. "If [students] can get an instant answer, does it add to their understanding or make it so they're just pushing buttons?" Mulcahy asks. "So many students are obsessed with calculators: 2+2 is 4 because that's what the calculator says, and if the calculator said otherwise, they'd go with that." He added, "Unless/until teachers — at all levels — teach and test more on the concepts (in addition to a certain level of computation), our students are doomed."
All these comments are based on an assumption that Wolfram Alpha will improve. It's already strong in math. The derivative of 5x is a fixed, universally agreed-upon quantity. And Wolfram Research has extensive experience doing such math with its Mathematica program, a fixture in many college courses.
The GDP of France, however, is continually updated and subject to revision. And the French government might decide tomorrow to add a column to the chart in its regular economic report, tripping up Wolfram Alpha's effort to peel the numbers out. "Wolfram Alpha seems to be quite poor at doing what it claims to do well — namely, adding value to something like a search engine by being able to carry out computations to generate data that's not already in place on someone's web page," said Jordan Ellenberg, a mathematician at the University of Wisconsin.
Jeff Witmer, who teaches statistics at Oberlin College, says he's already adjusted his curriculum to a more conceptual plane that wouldn't be majorly affected by this new tool. "I expect that Wolfram Alpha could be used to help answer questions that I asked on statistics exams 15 years ago, when I had my students do a lot of calculating," Witmer said. "These days I mostly ask students to interpret calculations that have already been made, which means that Wolfram Alpha would not be helpful."
Phil Hanser, a statistician with the Brattle Group, criticizes the search engine for not reporting its uncertainty about statistical calculations. "If Alpha is meant to be a pre-eminent mathematics search engine, it should also serve as a model of good mathematics practice, including statistics," Mr. Hanser says.
Other teachers noted that they'd already been using Mathematica in their teaching. "There's not a lot new there, besides that it's free as long as you have Internet access, which is not a small thing," said Dan Teague, who teaches math at the North Carolina School of Science and Mathematics. (Surprisingly, Wolfram said sales of Mathematica have increased since the launch of the new tool, thanks to increased exposure of his signature software package.)
What do you think? How will this new tool affect teaching? How will you use it, if at all? What searches make it stumble
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With the Discovering Geometry online textbook, students will navigate through investigations, explore dynamic figures, solve problems, and enrich their mathematical understanding—all from their desktop! They can leave their backpacks at home and their textbooks at school, and still continue their mathematical journey. The Discovering Geometry online textbook includes all the content of your classroom text.
Check out these enhanced features:
Easy navigation: Click a lesson title in the left margin to skip to that location. Or click the handy navigation buttons running across the bottom of each page.
Interactive glossary terms: As you study your Discovering Geometry online textbook you'll see blue, bolded words here and there, indicating terms that are defined in the glossary. Instead of flipping to the back of the book, just hover your cursor over the blue term and its glossary term will appear.
Hints for Selected Exercises: Click the icon and you'll jump to the hint for that exercise in the "Hints for Selected Exercises" section at the back of the book.
Interactive index terms: Look up a subject in the index, but instead of slowly navigating all the way to its referenced page, just click the page number following the index entry and you'll jump to the desired page.
"Practice Your Skills" links: When you want more practice with the basic skills of a lesson, click the "Practice Your Skills" icon. Your Web browser will go to a page containing additional problems for that lesson that you can download.
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Related Photo:
Since beginning my career as a teacher of mathematics almost 42 years ago, I have rarely heard a parent or student ever note Algebra as his/her favorite subject. Nevertheless, algebra skills do come in handy in many everyday situations.
The concept of algebra must be fully understood before you can gain full knowledge of mathematics. It becomes more advanced as you progress in school, thus it is essential for you to get a firm grip on all the skills at the most basic level before you move on to the next level.
Algebra must be taken one step at a time. If you do these steps over and over, your brain will make the adjustment and you may even begin to like it. The only way to ever learn algebra is through practicing its concepts and principles repeatedly in order to learn them. Yes – practice, practice, practice!
Practice, however, does not make perfect. Perfect practice makes perfect. Conceding that no one is perfect, effective practice is what you need to strive for. One effective method of practice is to use your text book so you can transfer all the concepts (along with equations that represent that concept) to a notebook. Only place one concept on each page so you can make notes and write done any ideas that help you remember how to work them correctly.
There are many books, online courses and tutors to assist you in breaking down algebra into its easiest components. As a rule, it is easier to learn for a person already familiar with algebraic concepts. You may also want to search out online communities where people discuss their algebra problems and assist each other in solving them.
One of the issues that make Algebra so difficult is the many different concepts and equations that must be conquered. You will not – and cannot – be successful unless you understand how all those work together (known as the entirety of algebra); you just will not be able to determine which procedure to use.
One of the key tools in algebra is a concept you learned in elementary school, order of operations. PEMDAS is an extremely valuable tool because you must understand which process happens first and so on. If you do not understand the common basics of algebra then you will be lost forever when dealing with this subject.
Your comfort will increase the more confident you become – then you can strengthen your skills by applying simple algebraic applications to your everyday life. By doing so, you will find that algebra is not quite as difficult as you may have once believed
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Differential Equations: An Introduction to Modern Methods and Applications
Unlike other texts in the market, this second edition presents differential equations consistent with the way scientists and engineers use modern methods in their work. Technology is used freely, with more emphasis on modeling, graphical representation, qualitative concepts, and geometric intuition than on theoretical issues. It also refers to larger-scale computations that computer algebra systems and DE solvers make possible. More exercises and examples involving working with data and devising the model provide scientists and engineers with the tools needed to model complex real-world
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Summary: A text for a precalculus course for students who have completed a course in intermediate algebra or high school algebra II, concentrating on topics essential for success in calculus, with an emphasis on depth of understanding rather that breadth of coverage. Linear, exponential, power, and periodic functions are introduced first, then polynomial and rational functions, with each function represented symbolically, numerically, graphically, and verbally. Contains many ...show moreworked examples and problems using real world data. Can be used with any technology for graphing functions.
From the Calculus Consortium based at Harvard University, this comprehensible book prepares readers for the study of calculus, presenting families of functions as models for change. These materials stress conceptual understanding and multiple ways of representing mathematical ideas. ...show less
2003 Paperback Fair 2nd
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Essential Math Methods F/phy.>intl Ed.<
9780120598786
ISBN:
0120598787
Publisher: Elsevier Science & Technology Books
Summary: This new, more accessible version of Arfken/Weber's blockbuster bestseller Mathematical Methods for Physicists 5/e is the most comprehensive and modern resource for using mathematics to solve physics problems. New features: * Many detailed, worked-out examples illustrate how to use and apply mathematical techniques to solve physics problems * Frequent and thorough explanations help readers understand, recall, and app...ly the theory * Introductions and review material provide context and extra support for key ideas * Many routine problems reinforce basic, foundational concepts and computations REVIEWERS SAY: "Examples are excellent. They cover a wide range of physics problems." - Bing Zhou, University of Michigan "The ideas are communicated very well and it is easy to understand...It has a more modern treatment than most, has a very complete range of topics and each is treated in sufficient detail....I'm not aware of another better book at this level..." -Gary Wysin, Kansas State University
Arfken, George B. is the author of Essential Math Methods F/phy.>intl Ed.<, published under ISBN 9780120598786 and 0120598787. Five Essential Math Methods F/phy.>intl Ed.<
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Advanced Mathematics for Engineers and Scientists - 71 edition
Summary: This Schaum's Study Guide is the perfect way for scientists and engineers to master the tools of advanced mathematics for scientists and engineers. Fully stocked with solved problemsN950 of themNit shows you how to solve problems that may not have been fully explained in class. Plus you get hundreds of additional problems to use for practice, with answers at the back of the book. Ideal for independent study, brushup before exams, or preparation for professional tests...show more, this Schaums Guide is clear, complete, and well-organized. It's the perfect supplement for any course in advanced mathematics for science and engineering, and a super helper for the math-challenged. This SchaumOs Outline provides a comprehensive review of advanced mathematical theory and methods youOll really use in high-tech industries and scientific
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Applied Math for Food Service - 98 edition
Summary: This book acquaints students with the basics of food cost controls by providing a foundation of practical techniques useful in real-world situations. Among these are yield tests, the calculation of recipe costs, and the use of food cost percentages. It further ensures that these future chefs are familiar with accurate measurements, portion control and proper food handling, which are essential to the healthy bottom line of any food service operation. Presents the mat...show morehematical skills necessary for food service professionals through a common sense, step-by-step approach using real life situations rather than algebraic formulas. This provides students with real-world applications for what they learn. Includes a detailed instructor's manual that assists the instructor plan classes, homework and quizzes; outlines lecture notes; provides additional student exercises; overhead transparencies; and solutions to all problems and questions. Provides extensive pedagogical aids throughout:
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Pages dog-eared. Soft cover may have some wear, especially along edges and/or may have some indentations, slight creasing or corner curl, though the text pages are good to very good. GOOD with average...show more wear to cover and pages. We offer a no-hassle guarantee on all our items. Orders generally ship by the next business day. Default Text. ...show less
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01-01-1997 Softcover Very good
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second edition aligned for the 2008-2009 testing cycle, with a full index. REA's test prep for the FCAT Math Grade 10 provides all the instruction and practice students need to excel on this state-required math exam. The book's comprehensive review features numerous examples, diagrams, and charts that illustrate and reinforce key math topics on the exam, including: Number Sense, Concepts, and Operations; Measurement; Geometry and Spatial Sense; Agebraic Thinking: and Data Analysis and Probability. Features 2 full-length practice tests so you can familiarize yourself with the content and format of the actual FCAT examDETAILS:- Fully aligned with the official state exam- Comprehensive review of every math topic on the exam- 2 full-length practice tests plus answer key and detailed explanations- - Packed with proven test tips, strategies and drills- Student-friendly classroom/homework assignments that reinforce key concepts REA...Real review, Real practice, Real results
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Trigonometry - 2nd edition
Summary: Engineers trying to learn trigonometry may think they understand a concept but then are unable to apply that understanding when they attempt to complete exercises. This innovative book helps them overcome common barriers to learning the concepts and builds confidence in their ability to do mathematics. The second edition presents new sections on modeling at the end of each chapter as well as new material on Limits and Early Functions. Numerous Parallel Words and Math...show more examples are included that provide more detailed annotations using everyday language. Your Turn exercises reinforce concepts and allow readers to see the connection between the problems and examples. Catch the Mistake exercises also enable them to review answers and find errors in the given solutions. This approach gives them the skills to understand and apply trigonometry. ...show less
LOOSE LEAF VERSIONPlease read description before purchase>> annotated teacher edition with publisher notation review copy not for sale on cover with all Students content and all solutions/ answers. te...show morext only no access code or other supplements. loose leaf needs binder (not included) ship immediately - Expedited shipping available ...show less
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Book Description: Consumer Mathematics presents basic math skills used in everyday situations—paying taxes, buying food, banking and investing, and managing a household. The full-color text helps learners of all ages become wiser, and more informed.
Consumer Mathematics student workbook
Book Description: Consumer Mathematics presents basic math skills used in everyday situations—paying taxes, buying food, banking and investing, and managing a household. The full-color text helps learners of all ages become wiser, and more informed
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MATLAB Student Version
04/01/03
Students in engineering, math or science have a new technical computing resource designed for their needs. The MathWorks' MATLAB Student Version includes full-featured versions of MATLAB and Simulink, the software products used by engineers, scientists and mathematicians at leading universities, research labs, technology companies and government labs. MATLAB integrates computation, data analysis, visualization and programming in one environment. Simulink is one of the leading interactive environments for modeling, simulating and analyzing dynamic systems. In addition, there is no difference between the student and professional versions of the program, which, according to the company, is important because students are learning skills with the same tools they may use in a professional arena. The program also comes with MATLAB and Simulink books to help students get started. This product has a special student price of $99. The MathWorks, (508) 647-7000
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9780321046246We're Sorry Not Available
More New and Used from Private Sellers
Starting at $6115College Geometry : A Discovery Approach
Summary
College Geometryis an approachable text, covering both Euclidean and Non-Euclidean geometry. This text is directed at the one semester course at the college level, for both pure mathematics majors and prospective teachers. A primary focus is on student participation, which is promoted in two ways: (1) Each section of the book contains one or two units, called M oments for Discovery,that use drawing, computational, or reasoning experiments to guide students to an often surprising conclusion related to section concepts; and (2) More than 650 problems were carefully designed to maintain student interest.
| 677.169 | 1 |
Unit 2: Linear Equations and Inequalities 5 weeks
Unit Overview
Essential Questions:
What is an equation?
What does equality mean?
What is an inequality?
How can we use linear equations and linear inequalities to solve real world problems?
What is a solution set for a linear equation or linear inequality?
How can models and technology aid in the solving of linear equations and linear inequalities?
Enduring Understandings:
To obtain a solution to an equation, no matter how complex, always involves the process of undoing the
operations.
UNIT CONTENTS
Note: The bolded Investigations are model investigations for this unit.
Investigation 1: Undoing Operations (1 day)
Investigation 2: Unit Pre-Test and Review (3 days)
Investigation 3: Two-Step Linear Equations in Context (5 days)
Investigation 4: Two-Step Linear Inequalities (2 days)
Mid-Unit Test (1 day)
Investigation 5: Multi-Step Linear Equations and Linear Inequalities (8 days)
Performance Task: iPods (3 days)
End-of- Unit Test (2 days including review)
Appendices: Investigation 2: Unit Pre-Test, Activity 2.1a Student Worksheet, Activity 2.1b Student
Worksheet, Activity 2.1c Student Worksheet Mid-Unit Test, Unit 2 Performance Task,
End-of-Unit Test.
Course Level Expectations
What students are expected to know and be able to do as a result of the unit
1.2.1 Develop and apply linear equations and inequalities that model real-world situations.
1.3.1 Simplify and solve equations and inequalities.
2.1.1 Compare, locate, label and order integers, rational numbers and real numbers on number lines, scales
and graphs.
2.2.1 Use algebraic properties, including associative, commutative and distributive, inverse and order of
operations to simplify computations with real numbers and simplify expressions.
2.2.2 Use technological tools such as spreadsheets, probes, algebra systems and graphing utilities to organize,
analyze and evaluate large amounts of numerical information.
2.2.3 Choose from among a variety of strategies to estimate and find values of formulas, functions and roots.
2.2.4 Judge the reasonableness of estimations, computations, and predictions.
3.3.1 Select and use appropriate units, scales, degree of precision to measure length, angle, area, and volume
of geometric models.
Vocabulary
algebraic expression distributive property literal equations
associative property evaluate order of operations
coefficient integers properties of equality
constant inverse operations real numbers
cummutative property linear inequalities simplify
variable
Assessment Strategies
Performance Task(s) Other Evidence
Authentic application in new context Formative and Summative assessments
CT Algebra One for All Page 1 of 23
Unit Plan 2, 8 13 09
iPODS Warm-ups, class activities, exit slips, and homework
Students will work on a two-day task that has them have been incorporated throughout the investigations.
investigating file storage size and cost for various In addition, the students will take a Unit Pre-Test,
models of iPods™. Students will share their findings Mid-Unit Test, and Formal End-of-Unit Test.
with the class.
INVESTIGATION 1 – Undoing Operations (1 day)
Students will discover the underlying algebra and see why number puzzles always work. In the process, they
will represent real world situations with algebraic expressions, evaluate an algebraic expression for a given
value of a variable, apply real number properties to simplify algebraic expressions, and describe reasonable
values that the variable and/or expression may represent.
Suggested Activities
1.1 You may use a number puzzle such as the one below to launch the lesson. Have the students perform the
following steps on a calculator or by hand:
Step 1: Enter the number of the month in which you were born
Step 2: Multiply by 5.
Step 3: Add 6
Step 4: Multiply the sum by 4.
Step 5: Add 9
Step 6: Multiply the sum by 5.
Step 7: Add the number of the day you were born.
Step 8: Subtract 165.
Ask the students, "What do you notice? Is your observation true for everyone in class?" The result should
be the month and day of their birthday. Ask the students, "Why did this happen? Can you prove that the
month and day of your birthday will always be the result when performing these steps? How?" Consider
having the students work in pairs or small groups to try to figure out how to justify this result. Have the
students share their work with the class. Emphasize the idea of "undoing" the steps. This, along with the
concept of a variable, will be an extremely vital concept throughout this unit.
An alternative puzzle to use is a simpler version for students who would benefit from starting with a less
complex problem. The solution to this puzzle will always be 5.
Step 1: Pick any number
Step 2: Take the number you picked and double it
Step 3: Add 10
Step 4: Divide the new number in half
Step 5: Subtract your original number
1.2 You may choose to have the students create their own number puzzles and try them out on each other. Or,
you may choose to use the website which has a nine-step number puzzle which,
no matter what number you start with, will always result in the number 350. This number is the "red line"
for humans, the carbon footprint that proponents think is necessary to sustain the earth. A video about
carbon footprints can be seen at: .
Assessment
By the end of this investigation students should be able to:
represent real world situations with algebraic expressions;
evaluate an algebraic expression for a given value of a variable;
apply real number properties to simplify algebraic expressions; and
describe reasonable values that the variable and/or expression may represent.
CT Algebra One for All Page 2 of 23
Unit Plan 2, 8 13 09
INVESTIGATION 2 - Unit Pre-Test and Review (3 days)
Students will take a pre-test on the prerequisite skills required for success in this unit. You may use the results
to plan any needed review or adjust the amount of instructional time needed as students work on the activities in
this unit.
Suggested Activities
2.1 Administer the Unit Pre-Test, particularly if you have noticed that some students had difficulty writing the
expressions that described the explicit rules in Unit 1.
You may find from the results of this diagnostic assessment that some students may require more review of
prerequisite skills and concepts than other students. These skills include work with integers, order of operations,
evaluating expressions, simplifying expressions, the commutative, associative and distributive properties,
solving one-step equations, using properties of equality, solving simple literal equations and solving word
problems that involve one-step equations.
You may use some of the following ideas to keep those students who do not require an extensive review
challenged and engaged in relevant mathematics while you plan and implement a review of the concepts
identified as weaknesses with the remainder of the class. Note: If you recognize a weakness in solving two-step
linear equations, then do not review this topic here. The next investigation in this unit focuses on solving two-
step linear equations.
You may choose to have students who are ready for enrichment work individually, in pairs, or in small groups
to complete one or more activities. One option is Activity 2.1a Student Handout - New Cell Phone Plan.
After students complete the worksheet they may research (either via the web or the newspaper) various cell
phone plans. They might then use the data to compare plans and select the best buy.
Another option is to have students work individually, in pairs, or in small groups and complete either the
standard Activity 2.1b Student Handout - Recycling Activity or the differentiated Activity 2.1c Student
Handout - Recycling Activity. The differentiated version includes some more challenging questions.
Assessment
By the end of this investigation students should be able to:
perform integer operations;
combine like terms;
evaluate expressions;
use the distributive property;
solve one-step linear equations;
solve one-step linear inequalities; and
solve one-step linear equations and inequalities in context.
INVESTIGATION 3 – Two-Step Linear Equations in Context (5 days)
Students will be able to write a linear equation that models a real world scenario, solve two-step linear
equations, and justify their steps.
See Model Investigation 3.
INVESTIGATION 4 – Two-Step Linear Inequalities (2 days)
Students will be able to write a linear inequality that models a real world scenario, solve two-step linear
inequalities and justify their steps.
Suggested Activities
4.1 Students may have worked with inequalities and solved linear inequalities in one step. However, when
CT Algebra One for All Page 3 of 23
Unit Plan 2, 8 13 09
dividing or multiplying both sides of an inequality by a negative, some students do not understand why you
have to change the direction of the inequality. If students have not developed a understanding of this prior
to this course, then the teacher may choose how to teach this concept. One suggestion is to use the TI-84
program LINEQUA (This program can be found at
and typing 8773 into the search box.
Or, you can download ideas about how to use this program in your classroom.). Another suggestion is to use
the more traditional approach of testing values for the variable in the original inequality and the simplified
inequality (if they are equivalent, then the solution set should be the same).
4.2 To begin work with inequalities you might start with a problem of interest and practical use to students. For
example, students may go to a specific website such as, and look up the cost of a
popular item that might be a good item for a fundraiser or sports booster sale. They might pick out the shirt
they think everyone should wear at the annual pep rally. Have students find the cost of one of the shirts.
Suppose the student council has set aside $6,000 to purchase the shirts. (They plan to sell them later at
double the price.) Then ask them to determine how many shirts they can buy at the price they found online
if the shipping costs are $14. At this point, you might want to remind students that schools are tax exempt.
Students might want to set up an equation at this point. You can have a discussion with the students about
how you don't have to spend all $6,000 dollars but you definitely can't spend more than that. If the T-shirts
are $21.96 each, then the inequality would be $21.96x + 14.00 $6000, where x is the number of T-shirts.
Be sure that students define the variable. You can then have students explain how they would solve for x
and have them show their steps and check their answer. In this case, x is about 272.6. Facilitate a discussion
with the class regarding this answer and why the solution shows that they can buy up to 272 T-shirts. Have
them discuss why rounding down in this case is necessary versus rounding to the closer value of 273 shirts.
For differentiation, you can solve the equation $21.96x +14.00 = $6,000 and then try to solve the inequality
again using the equation as a model. Students should also use a number line to get a visual of the solution
set.
For more practice, solve similar problems after finding the costs for uniform shirts for the football team,
baseball hats, or something else of interest to students.
An alternative problem would be to have students visit a cellular phone company website and find a plan
that they think fits the needs of their family based on the number of minutes they use a cell phone per
month. Then have them pick out the texting option that they think would fit the number of texts they send
per month. Lastly, determine if they need to add Internet as an option also. With this data, you can have
students set up an inequality that will help them to determine the number of months they can afford the cell
phone. For example, a cell phone family plan has 1,400 minutes for $89.99/month for the first two lines and
every line after that is $9.99, unlimited texting messages for $30.00/month, and Internet for $10.00/month
per line. Suppose the family has allotted $200.00 per month for cell phone lines for the family. Have
students write and solve an inequality to determine the number of cell phone lines that the family can have
with a $200.00 allotment. Remind them that they must define the variable. Possible solution: 200 89.99
+ 9.99x + 30 +10(x + 2), where x is the number of additional lines to the plan. In this case, 3 additional
lines, for a total of 5 lines, can be on the plan.
As an extension, students can also go back and calculate how much it would cost each individual member of
the family to have an equivalent individual cell phone plan. Students can then determine how much a
family would possibly save by getting a family plan versus an individual plan for each family member.
For differentiation, you can have more advanced students also write and solve the inequality to include tax.
For students who are struggling with this concept, you can remove the fact that the $89.99 covers the first
CT Algebra One for All Page 4 of 23
Unit Plan 2, 8 13 09
two people. Instead make it so that $90.00 is the base price and every member costs $10.00/month. For
students who are still struggling, it may be helpful to use play money to act out the algebraic steps.
Also, for students who are really having a difficult time, you may want to find a couple of cellular plans
ahead of time and ask them simpler questions about the plans. This way, they will be dealing with the same
context, but won't have to contend with some of the trickier constraints.
Assessment
By the end of this investigation students should be able to:
Write and solve two-step linear inequalities in context;
justify why you flip the inequality sign when multiplying or dividing by a negative number; and
justify the steps in solving linear inequalities.
Mid-Unit Test (2 days)
INVESTIGATION 5 – Multi-Step Linear Equations and Linear Inequalities (8 days)
Students will be able to write a linear equation that models a real world scenario, solve multi-step linear
equations and linear inequalities, and justify their steps.
See Model Investigation 5.
Unit 2 PERFORMANCE TASK – iPods
iPods provides students an opportunity to apply what they have learned in Unit 2. They will investigate how
linear equations and inequalities can be used to help them make decisions. (See Unit 2 Performance Task-
Sample Student Handout.) Students may work in pairs or small groups during this performance task. Each
group will share their findings and recommendations with the class.
Suggested Activities
In a whole class discussion, let students know that they will assume the role of a writer for a local newspaper
who is investigating information about iPods. The reporter wants to determine whether or not the data is
accurate. His goal is to make recommendations to the marketers and consumers of this technology. Students
may work in pairs or small groups over the next two class periods to complete a variety of contextual problems
and share findings with the class. You may use a third day if students need more time to discuss ideas and write
their group's responses. After all groups have reported, you may have students assess the response of their
group and suggest ways that they may improve their responses.
As extensions to the activity sheet, students may act like reporters and keep a list of additional questions they
have as they do their "research" on iPods. Then, challenge them to find the solutions.
Students may write a newspaper column highlighting their findings. This would appeal to students who enjoy
creative writing. It might serve as a way to get them excited about mathematics as a vehicle for narrative or
persuasive writing.
Yet another extension is to have the students create a newspaper advertisement for the 16GB iPod Nano or the
1TB iPod to accompany the newspaper column. Students should include the relevant solutions they found as
they create the advertisement. Going even further, the advertisement shared with the class as a PowerPoint
presentation or an audio or video commercial. The presentations should include references, sources of their
data, and important data and calculations.
Students may notice that the cost of iPods is going down and that storage size and performance are increasing.
They may wish to do an internet search of recent models and costs. Their search might start with Moore's Law.
End-of-Unit Test (1 day)
CT Algebra One for All Page 5 of 23
Unit Plan 2, 8 13 09
Technology/Materials/Resources/Bibliography
Technology:
Classroom set of graphing calculators
Graphing software
Whole-class display for the graphing calculator
Computer
Overhead projector with view screen or computer emulator software that can be projected to whole
class, and interactive whiteboard
On-line Resources:
NUMB3RS "Burn Rate" Activity:
350 Number puzzle:
350 Video:
Information on Electoral College:
Information on Presidential Pets:
Mathematics and the Police:
IRS:
Create a class blog page at:
LINEQUA: (Search 8773)
Look-up your school colors here:
Body Mass Index (BMI) info:
Video from NUMB3RS TV show:
NUMB3RS "Burn Rate" activity:
Poll Everywhere site:
Build a backyard regulation-size volleyball court:
Cool theaters:
Materials:
Algebra Tiles
Coins or counters
Bibliography: (Helpful print resources)
Horak, V.M. (2005). Biology as a Source for Algebra Equations: The Heart. Mathematics Teacher: 99(4).
Horak, V.M. (2005). Biology as a Source for Algebra Equations: Insects. Mathematics Teacher: 99(1).
Kunkel, P., Chanan, S., & Steketee, S. (2006). Exploring Algebra 1 with The Geometer's Sketchpad. CA: Key
Curriculum Press.
CT Algebra One for All Page 6 of 23
Unit Plan 2, 8 13 09
UNIT 2 PRE-TEST
NAME ________________________________________________ DATE ____________________
Directions: This pre-test is designed to find out how much you know about several math topics that will be part
of the algebra course. Try and answer every problem on this pre-test and show the work you did to complete the
problem.
1. Calculate the value of each math expression below:
a. 3 + 2 • 4 ____________ b. 5 – 2(9 – 5) _________
c. 4 • 9 – 12 ÷ 6 __________ d. 5 •32 + 3 • 3 +10 __________
2. Find the answer for each of the problems below:
a. 8 + (- 5) = _______ b. (- 3) – (- 5) = ______
c. 14 ÷ (- 2) = _______ d. (- 6) • (- 7) = _______
3. If water is pouring into a tank at the rate of 15 gallons every 4 seconds, how long will
it take to completely fill a tank that holds 900 gallons? ______________ Show your work.
4. Solve each of the following equations for x. Show your work.
a. x + 12 = 23 _____ b. x – 6 = 34 ________
x
c. 6x = 38 _______ d. 7
4
5. Solve each of the following equations. Show your work.
x
a. 3x + 5 = 20 _________ b. 7 2 __________
3
CT Algebra One for All Page 7 of 23
Unit Plan 2, 8 13 09
6. Your high school had a tag sale to raise funds for a local charity. It cost five dollars to get in and three
dollars for every item you purchased. If you spent twenty-three dollars, how many items did you buy?
Write an equation that models the problem, and then solve it. Show your work.
7. Use the formula M = 5T + 3 to fill in the values of M in the chart, given the
values of T.
T M
1
2
3
4
5
8. a. Give an example of the Commutative Property of Addition:
b. Give an example of the Associative Property of Multiplication: __________
9. Use the Distributive Property to rewrite the following expression without parentheses:
3(4x – 8) = ____________
10. Simplify: 4x + 7 + 3x – 10 + x
11. Solve each equation for the designated variable:
a. d = rt b. P = a + b + c
Solve for t. ____________ Solve for b. ____________
CT Algebra One for All Page 8 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1a, p. 1 of 2
New Cell Phone Plan
Name: ___________________________________________ Date: ______________________
Suppose you are shopping for a new cell phone plan. The table represents the various plans that you can
purchase from a local communications store.
x = # of minutes Cost of plan
Plan
used per month per month
A x< 450 $39.99
B 450 x < 900 $59.99
C x 1350 $79.99
D Unlimited $99.99
a. Describe two different months of possible minutes that someone could use in Plan A and not be charged
overage fees.
b. Would all numbers less than 450 be possible under plan A?
c. Could a customer use exactly 450 minutes in plan A and pay $39.99?
d. If we were to graph all of the possible values that would work under plan A, how many points would
you have to plot?
e. On the number line, shade all of the possible minutes that could be used in plan A without being charged
overage fees. Be sure to label points of reference on your number line.
Plan A Possible Minutes
f. On the number line below, graph all of the possible minutes for plan B.
Plan B Possible Minutes
CT Algebra One for All Page 9 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1a, p. 2 of 2
g. On the number line below, graph all of the possible minutes for plan C.
Plan C Possible Minutes
h. On the number line below, graph all of the possible minutes for plan D.
Plan D Possible Minutes
i. You need to choose a plan. During the school year you use your cell phone less frequently than in the
summer. You estimate that, on average, you use about 800 minutes per month. But in June, July and
August you use about 1200 minutes per month.
For any of these plans, you have to pay $60.00 extra for any month in which you exceed the plan's limit.
Which plan should you choose if you have to sign up for a one-year contract?
Show your work.
CT Algebra One for All Page 10 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1b, p. 1 of 1
Recycling
Name: ___________________________________________ Date: ________________________
You are collecting aluminum cans to raise funds for a local dog shelter. Should you bring the cans to the
supermarket or the recycling center?
If you bring the cans to the supermarket, you receive 5 cents per can as a return on the deposit.
If you bring the cans to the recycling center, you receive 6 cents per can but also must pay a flat $15 recycling
fee.
Suppose you collect 5000 cans.
a. How much would you get if you took the cans to the supermarket?
b. How much would you get if you took the cans to the recycling center?
c. If you took some cans to the supermarket and got $6.50, how many cans did you take?
d. Write an equation to solve c.
e. What does the variable you used represent?
f. Solve the equation.
g. Write an equation to find the number of cans you brought to the recycling center if you got $18.60?
Solve the equation.
h. How many cans would you have to collect in order to receive the same amount at the recycling center or
the supermarket?
i. Describe how you found your answer.
CT Algebra One for All Page 11 of 23
Unit Plan 2, 8 13 09
Unit 2, Investigation 2
Activity 2.1c, p. 1 of 1
Recycling
Name: ____________________________________________ Date: ___________________________
You are collecting aluminum cans for a fund-raising drive. Should you bring the cans to the supermarket or the
recycling center?
If you bring the cans to the supermarket, you receive 5 cents (.05) per can as a return on the deposit.
If you bring the cans to the recycling center, how much will you receive?
They pay a base amount of 4 (.04) cents per can.
Plus an additional 0.3 (.003) cents per can to 1.2 (.012) cent per can depending on the current value of
scrap metal.
Plus a bonus of 0.35 (.0035) cents per can for every can over 1000 cans.
They charge a flat fee of $7.50 for each lot of cans brought in (because small lots require nearly as much
handling as large lots).
Suppose you collect 10,000 cans.
a. How much money would you need to receive from the recycler in order to make the trip equal to what
you could get at the local supermarket? Explain your thinking.
b. Assign a variable to the current scrap metal value per can. _________________________
c. Write an expression containing the variable which tells the money received from the recycler for 10,000
cans.
d. How much would you get if you took the cans to the supermarket? __________________
e. Use your answers to problems c. and d. to write an equation. Solve the equation for the variable.
CT Algebra One for All Page 12 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout, p. 1 of 3
iPods!
Name: ________________________________________________ Date: ______________________
As a part-time job, you write the Consumer Watch column for your local newspaper. This week's column is
going to be on the 16 GB iPod nano.
Someone wrote in to your column and attached a printout from a website that claimed that the 16 GB iPod nano
has "16 GB capacity for 4,000 songs or 16 hours of video." This seemed interesting to you so you decided to
investigate.
You found this information:
Downloading Downloading iPod nano iPod touch
songs on iTunes: movies on iTunes: 8 GB - $149 8 GB - $229
Each song costs Each movie costs 16 GB - $199 16 GB - $299
$0.99 $14.99 32 GB - $399
and takes up and takes up
approximately 5 MB approximately 1.5 GB
You also did some research and found that 1,024 MB = 1 GB.
For each of the following questions, be sure to define the variables you used in each question and show your
work.
1. If the website's claim for the songs is accurate, how large (in MB) is the average song? How does this
compare to the data you found?
2. Check information from the site: (or any
other site you find) to see if the claim in the ad seems reasonable. Another site you may want to investigate
is:
3. It seems reasonable that the length (L) of a song is related to its size in MB (S). That is, the longer a song,
the larger its size in MB should be. Use the website information from question #2, or another site you find,
to develop a formula that relates L and S.
4. Assuming you can fit 4,000 songs on the 16 GB iPod nano, how much would it cost to buy it and fill it with
songs? Justify your answer.
CT Algebra One for All Page 13 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout p. 2 of 3
5. How long would it take you in days and hours to listen to all 4,000 songs? Round to the nearest hour if
appropriate. Also, be sure to carefully explain your reasoning.
6. One of the perks of the job is that you often get to test the products you are investigating. Your editor has
given you an iPod and $250 to purchase songs and videos to test it out. You want to give your readers some
idea of their downloading options. You assume that most people have, on average, 45 times more songs than
they do videos. Complete the following chart:
# of Videos # of Songs Total Cost Total Size of Files
1
2
3
4
5
7. Let us assume that you bought a 16 GB iPod nano and downloaded two movies for it. Construct an
inequality to determine the number of songs that will fit on it. As always, be sure to define your variables
and solve the inequality.
8. You hear that Apple is planning to introduce a 24 GB iPod nano this summer. What price do you estimate it
will sell for? Be sure to explain your reasoning.
9. The ad that you found also mentioned that a fully charged 16 GB nano can play "Up to 24 hours of music
when fully charged." About how many songs can you play before it runs out of power?
CT Algebra One for All Page 14 of 23
Unit Plan 2, 8 13 09
Unit 2 Performance Task
Sample Student Handout p. 3 of 3
10. Since computers are continually getting smaller and smaller and can hold more and more information, there
is a possibility that one day there will be a one terabyte (TB) iPod. Find out how large a terabyte is and use
that information to estimate what it would cost to load a one-TB iPod with movies.
11. An important aspect of iPods is their ability to play both audio and video podcasts. For example, check out
the podcast on Muslim Contributions to Mathematics:
In it, there is mention of al-Khawarizmi, often considered the Father of Algebra. He named the study aljabr,
which obviously sounds very similar to the word algebra.
Because of all the numbers and variables involved in exploring the iPod, it would be very helpful for your
readers if they had somewhere to go to "brush up" on the algebra needed to understand your column. Find a
podcast that helps people understand variables or equations. List the website or the source of the podcast
and write a brief description of the podcast.
Note: A website that is helpful for product reviews is
CT Algebra One for All Page 15 of 23
Unit Plan 2, 8 13 09
UNIT 2 MID-TEST
After Investigation 3
EQUATIONS
NAME ________________________________ DATE ________
1. Simplify the following expressions by combining like terms. Show all work.
a. (3x – 7) + (x + 9) b. 12x – (4x – 3)
c. 3(2x + 5) + 4(x – 8) d. 2x – 5(2x + 1)
2. Evaluate each expression if x = 5. Show all work.
1
a. 14 – 4x ÷ 10 b. –(6 – 12) ÷ 3 + 10( )
x
3. Solve the following equations. Show your work and state which Property of Equality you used.
x
a. x + 3 = -2 b. 5x = 32 c. 6
7
4. Solve the following inequalities. Show work and show solutions on the given number lines.
a. x – 2 > 5 b. 5x 15
5. Pedro and Janelle are given the task by their families of getting the rubbish ready for pickup each week
during the summer vacation. Pedro is paid $4 per week while Janelle is paid $15 at the beginning of the
summer and $2 each week. Choose variables and explain what they represent.
CT Algebra One for All Page 16 of 23
Unit Plan 2, 8 13 09
Write an equation for each situation.
Pedro: ________________________ Janelle: ___________________________
Which method of payment would you prefer? Explain your choice.
6. A store could use the equation C = 7.50 + 1.75w to calculate the price C it charges to ship
merchandise that weighs w pounds.
a. Find the price of mailing a 5-lb package. ______________
b. What is the real world meaning of 7.50?
c. What is the real world meaning of 1.75?
7. Solve the equation, 45 = 4c – 11, for c. Explain what Property of Equality was used in each step in your
solution.
45 = 4c - 11
8. Oscar and his band want to record and sell CDs. There will be a set-up fee of $400, and each CD will cost
$3.75 to burn. The recording studio requires bands to make a minimum purchase of $1000.
a. Write an equation that relating the total cost to the number of CDs burned.
______________________________
b. Write and solve an inequality to determine the minimum number of CD's the band can burn to meet
the minimum purchase of $1000.
9. Solve the following equations. Show your work.
x
a. – 6 – 9x = 3 b. –3x + 10 = 10 c. 13 15
6
CT Algebra One for All Page 17 of 23
Unit Plan 2, 8 13 09
10. Solve the following inequalities. Show your work.
a. –2x – 10 < -8 b. 14 10x-26
11. At the Berkshire Balloon Festival, a hot air balloon is sighted at an altitude of 400 feet and appears to be
descending at a rate of 25 feet per minute.
a. Write an equation that relates the height of the balloon to the rate the balloon is descending per minute.
b. How high is the balloon 10 minutes after it is spotted at 400 feet? Show your work.
c. How high is the balloon 5 minutes before it is spotted at 400 feet? Show your work.
d. How long will it take the balloon to land after it is spotted at 400 feet? Show you work.
CT Algebra One for All Page 18 of 23
Unit Plan 2, 8 13 09
UNIT 2 TEST
EQUATIONS
NAME: ________________________________ DATE: ____________
1. The University of Connecticut (UCONN) defeated Louisville University in the finals of the NCAA
Women's Basketball Tournament in 2009 by a score of 76 to 54.
a. There are three ways to score points in college basketball:
score a basket within 20 feet (actually 20 feet 9 inches) for two points
score a basket farther than 20 feet for three points,
or score a foul shot for one point
Assign a variable for each scoring method.
b. Write an equation for determining the final score, F, for a team if you know the number of two-point
baskets, three-point baskets and foul shots.
c. Suppose UCONN scored 25 two-point baskets and 14 foul shots. Write an equation which will allow
you to find the number of three-point baskets the team scored.
Solve the equation.
d. In their last game, Louisville scored a total of 54 points, of which 42 points were foul shots or 2-point
baskets. In the equation, 54 = 3g + 42, g represents the number of three-point baskets Louisville scored.
Solve the equation for g.
CT Algebra One for All Page 19 of 23
Unit Plan 2, 8 13 09
2. The Bridgeport Bluefish management discovered that the profit they make on their concession stands may
be found by using the formula: P = 4V – 300, where P represents the profit and V represents the attendance
at the game.
a. Find the profit if the attendance is 1,200 fans. Show your work.
b. Find the profit if the attendance is 3,500 fans. Show your work.
c. What does the coefficient 4 represent in the equation?
d. What is the real world meaning of the 300 in the equation?
3. Two different formulas used to find the area of a trapezoid are below.
Show that the expressions in the formulas are equivalent.
1 1 1
A= h(a+b) A= ah + bh
2 2 2
CT Algebra One for All Page 20 of 23
Unit Plan 2, 8 13 09
4. In Vermont the speed limit on some major highways is 75 miles per hour. To find the
fine a speeder has to pay when he travels over the speed limit, the State of Vermont uses the following
procedure:
• Take the speed of the car and subtract 75
• Multiply the difference by $45
a. Select a variable to represent the fine __________________ and a variable to represent the miles per
hour the speeder is traveling ________________________.
b. Write an equation that shows the relationship between the fine and the miles per hour the speeder is
traveling.
c. Explain what values the variable may have for the miles per hour
d. Find the fine if the rate of the speeder is 95 miles per hour.
e. Find how many miles per hour the speeder was traveling if the fine is $360.
CT Algebra One for All Page 21 of 23
Unit Plan 2, 8 13 09
5. Latisha is traveling to Australia to study aquatic life along the Great Barrier Reef. She is planning her trip
and needs to know how much money to bring with her. This table shows recent Australian-dollar
equivalents of various U.S.-dollar amounts.
Dollar Conversion Table
U.S. Dollars Australian Dollars
15 18.9982
25 31.6514
35 44.3120
50 63.2999
a. Use the data in the table show how you could use the data in the table to find the number of Australian
dollars you would get for one U.S. dollar.
b. Write an equation for finding the number of Australian dollars, y, equivalent to x U.S.
dollars.____________________________________
c. Use your equation to find how many Australian dollars Latisha would receive for $1,400 U.S. dollars.
d. At the end of her stay, Latisha has $180 Australian dollars. Use your equation to find how many U.S.
dollars she will receive in exchange.
6. Solve the following equations and inequalities. Show all your work. For the two inequalities, graph the
solutions on the given number line.
a. 3(2x – 7) + 2x = 51 b. 3x – 2(x – 5) = 10- x
c. 6x + 3 = 2x + 15 d. 2(x + 3) + 4 = 30
2 1
e. 2x + 3 3x – 5 f. (2x-3)+ (x 4) 11
3 2
CT Algebra One for All Page 22 of 23
Unit Plan 2, 8 13 09
7. This expression describes a number trick:
(6(N 3) 12)
N
6
a. Test the number trick with two different starting numbers. Show your work.
b. Did you get the same result for both numbers?
c. Tell which operations undo each other in the number trick.
CT Algebra One for All Page 23 of 23
Unit Plan 2, 8 13
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This lesson received an honorable mention in the 2014 SoftChalk Lesson Challenge.'We have seen quadratic functions which...
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A critical overview of the current debate and topical thinking on international comparative investigations in mathematics education. The contributors are all major figures in international comparisons in mathematics. The book highlights strengths and weaknesses in various systems worldwide, allowing teachers, researchers and academics to compare and contrast different approaches. A significant contribution to the international debate on standards in mathematics.
This book collects the work of thirty-five instructors who share their innovations and insights about teaching discrete mathematics. Whether you teach at the college or high school level; whether your students are from mathematics, computer science, or engineering; whether you emphasize logic, proof, counting, graph theory, or applications, you will find resources in this book to supplement your discrete mathematics course.
The very ancient Indian Mathematics Tricks. Vedic is the Holy Book of Hindu. Just one example: Suppose you want the square of 35 (any two digit square ending with 5) Multiply 3 (1st dig) x 4 (1st dig +1) = 12 write then 5 x 5=25 write one after another So, The answer is 1225.
This is the first strand in a brand new series of Developing Mathematics, developed to be fully in line with the government's Revised Primary Framework for Mathematics. It features completely new content, a fresh up-to-date look both inside and out and 64 pages of attractive ready-made handouts for busy teachers.
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After also…
Dear User, your publication has been rejected because WE DO NOT ACCEPT THIS SORT OF MATERIALS at englishtips.org. Please see our rules here: Thank you
Elementary
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ERRATACOMPUTER-AIDED ANALYSIS OF MECHANICAL SYSTEMSParviz E. NikraveshPrentice-Hall, 1988(Corrections as of January 1, 2004)Address to an error is given in the first column by the page number and in the secondcolumn by a line number, or a figure number, or an equation number. For example: "line2" means the second line from the top of the page; "line –3" means the third line from thebottom of the page; "Eq. 2.30, +2" means the second line following Eq. 2.30; Eq. 6.48,line 1 means the first line in Eq. 6.48.
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Geometry: Fundamental Concepts and Applications
9780321473318
ISBN:
0321473310
Edition: 1 Pub Date: 2007 Publisher: Pearson
Summary: This Geometry workbook makes the fundamental concepts of geometry accessible and interesting for college students and incorporates a variety of basic algebra skills in order to show the connection between Geometry and Algebra.
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because spreadsheets allow students to view the numerical representation of a
function beside the graphical display;
because most students have access to spreadsheet software.
When students are introduced to linear relations in grade 9, solve quadratic equations
in grade 10, and study exponential behaviour in grade 12 they often don't identify
connections between different function types. Each equation has its own mysterious
parameters. The behaviour of each graph is learned in isolation.
Through the following investigations, students work with a variety of functions,
altering parameters and observing the changes in the numerical data and in the graphical
representation. They then are asked to analyse the results, and to form conclusions
about functions in general.
Setting up Data and Plot windows
The spreadsheets in this unit were created using Microsoft Excel 5.0. They can be accessed
and used onscreen once your browser has been set up to use Excel as a Helper Application.
If you do not have Excel, please see alternate instructions
for setting up the activities.
Lessons
In the first part of this unit students explore the effects of altering parameter values
in a variety of function equations:
m in the equation y = mx b in the equation y = mx + b a, p and q in the equation y = a(x-p)2+q a and n in the equation y = axn
They examine the results in both the numerical data and the graph and predict what
changes will cause the graph to pass through particular points.
Access each spreadsheet and follow the instructions within.
Each sheet should have 3 windows - a table window, a plot window and an instruction window. If one is missing, use the UNHIDE command under the Window menu to view all three.
It may help, after reading the instructions, to close one window and enlarge the others. NOTE: Altering the spreadsheet, or saving it, will not affect the original!
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BARRON'S Math Workbook for the New SATARRON'S Math Workbook for the New SAT Description
Salient features Special math strategies to help you approach unfamiliar question types Intensive focus on third-year college preparatory math topics Hundreds of multiple-choice exercise questions with worked-out solutions "Grid-in" questions with worked-out solutions Arithmetic skills and concepts Algebraic methods Algebra II topics Word problems Geometric concepts and reasoning Special problem types Revised to reflect the newly structured math section on the SAT 2005 Effective March 2005, the math section of the SAT has been restructured and expanded in content. This thoroughly updated and comprehensive workbook offers test takers the best SAT math preparation available anywhere by keeping pace with all test changes. You shall find special math strategies to help you approach unfamiliar question types, as well as new content from third-year college preparatory math. Exercises include hundreds of multiple-choice and "grid-in" questions with worked-out solutions.
This special low-priced edition is for sale in India, Bangladesh, Bhutan, Maldives, Nepal, Myanmar and Sri Lanka only.
Contents: PART I: LEARNING ABOUT THE NEW SAT MATH Chapter 1: Knowing What You are Against
Similar Items by Category
Discussion : BARRON'S Math Workbook for the New SAT
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Free MATLAB and Simulink in Academia Interactive Kit
Learn why faculty, students, and researchers use MATLAB and Simulink across departments and applications in engineering, science, and math.
Get immediate access to this MATLAB and Simulink interactive kit by completing this form.
This interactive kit features popular webinars that demonstrate how MathWorks products can be used in a range of disciplines. More than 5000 academic institutions use MATLAB and Simulink for teaching and areas of research that require mathematical computation, data visualization, simulation, test, analysis, and more. Complete the form now to watch these webinars,read user stories, and access product data sheets.
Recorded Product Webinars
Demonstrations led by MATLAB and Simulink product experts
Origins of MATLAB
A Hands-On Approach to Teaching with LEGO MINDSTORMS and Simulink
Mathematical Modeling with MATLAB Products
Speeding Up MATLAB Applications
Introduction to Simulink
MATLAB for C/C++ Programmers
Physical Modeling with the Simscape Language
Image Processing with MATLAB
Introduction to Data Analysis with MATLAB for Life Scientists
Introduction to MATLAB
Learning Basic Mechatronic Concepts Using the Arduino Board and MATLAB
User Stories
RWTH Aachen University Prepares Students for Careers in Industry
Automotive Research Lab at Penn State Gives Students Practical Hardware-in-the-Loop Experience
MathWorks Tools Used to Predict Financial Crises in Emerging Markets
RTI International and University of Pennsylvania Model the Spread of Epidemics
Research Engineers Advance Design of the International Linear Collider
University of Waterloo Develops Award-Winning Fuel Cell Technology
University of Sydney Students Experience Flight Dynamics in the Variable Stability Flight Simulator
More than 1,000 Georgia Tech Engineering Students Learn Computer Science Concepts Each Semester with MATLAB
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Intermediate Algebra
9780321785046
ISBN:
0321785045
Edition: 6 Pub Date: 2011 Publisher: Pearson Education
Summary: Martin-Gay, K. Elayn is the author of Intermediate Algebra, published 2011 under ISBN 9780321785046 and 0321785045. Six hundred twenty five Intermediate Algebra textbooks are available for sale on ValoreBooks.com, two hundred forty four used from the cheapest price of $66.73, or buy new starting at $126.99.
Ships From:Multiple LocationsShipping:Standard, ExpeditedComments:Standalone book only-does not include online access card. Brand new in plastic. US edition as sh... [more]Standalone book only-does not include online access card. Brand new in plastic. US edition as shows. Will ship with tracking number. * examples that each chapter has throughout the book that breaks down different algebra problems and shows you step by step how to work certain formulas was extremely helpful.
The primary subject of this book is covering the basics of Algebra. I was required to use this text book for my Intermediate Algebra class and it was very effective in helping me work through different algebra problems.
this one is more specific and use a general vocabulary, any person can understand what to do. I've read books about this subject in high school. I like this one more than those.
it was for a computer class, we were supposed to learn how to use windows 8 and its applications. they put more attention in teach how to use Word, Excel, Access and Power Point. I liked so much because it explain step by step what you have to do, you don't feel lost doing your assignments. Also it helped me in other classes, some professors ask for presentation in power point or flyers in Word. so, I can do it without problem.
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Graphing for Less
It's accepted wisdom that students of High School math and advanced science classes need to use an expensive handheld graphing calculator. At $100 a pop, the price can add up to some serious cash for a school or district. There is an inexpensive, or free, alternative for schools where teachers and students have notebooks. They can be used for solving complex equations, graphing lab results and even working with pesky trig problems. The bonus is that with full-size color screens, notebooks can produce better looking results than even the best handheld calculator can. Here're four of my favorite calculator programs.
Think Holt, Rinehart and Winston and you think of textbooks, but the publisher also has a great Web-based graphing calculator available for a free download. It can work with up to four independent variables and has full trigonometric functions. On top of showing the intersections between lines, you can zoom in on any line.
The Java-based GCalc2 is incredibly powerful, but to make it work, you'll need t load the latest release of Java JRE software. It runs as an applet and presents a rather intimidating interface that definitely requires reading the manual to make the most of the program. Just type in your function in the unmarked text bar at the top and it's immediately graphed.
A more kid-friendly Java calculator comes from CoolMath's online math site, which, like the rest of the site, is a real winner. The center of attention of the Graphit Graphing Calculator is the keypad on the left, which has number keys as well as trig functions and the ability to zoom. On the downside, to graph a function, you need to hit the tiny Eval button, which is well hidden in the keypad.
The beauty of WebGraphic.com's online graphing calculator is that rather than a one size doesn't fit all approach, it has different levels for basic, intermediate and advanced work, making it perfect for different classes and needs. To use its most basic functions, you can just jump right in, but if you register, you can graph functions and solve systems of equations. Paying members get 3-D visuals as well as some truly cool math abilities, and it costs $75 for a class of up to 35 students and decreases as the volume rises. There are a bunch of how-to examples, but do yourself (and your class favor) and spend sometime exploring the Weird functions area.
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We conclude with a discussion of model characteristics and area of integer programming 91 symmetry an integer program is integer programming problems
We extend the technique for use in integer programming and mixed integer pro we explain why, and identify ematical programming system for non convex problems
Integer programming can be modeled as a later we shall explain why nonlinear programming is not attractive for for convex programming problems,
We conclude with a discussion of model characteristics and we explain how parameter settings solving mixed integer programming problems using automatic
Analysis of linear programming problems after the simplex the general characteristics of linear programming called integer programming
Use binary integer programming is this a general property of linear programming problems? explain why rounding or explain what characteristics the data
Georgia performance standards: investigate and explain the characteristics of a on the steps they took to solve linear programming problems in two
To use an integer programming formation of the problem this would be done by adding • linear programming problems constitute an important class of constrained
Solve linear programming problems using the graphical method 3 conduct and explain sensitivity have any values including no integer values that satisfy
Linear programming may be used to solve a relaxation of an integer program the problems below represent binary integer characteristics the final solution
Investigate and explain characteristics of all integer exponents programming problems in two variables a
142 apply the branch and bound method to solve integer programming problems 154 analyze by computer a variety of operating characteristics of explain how
141 apply the branch and bound method to solve integer programming problems 154 analyze by computer a variety of operating characteristics of explain how
Comparing and contrasting characteristics of a variety of functions in order use "linear programming if it is not linear have them explain why
Problems; they are hungry and "introduction to programming" some of its characteristics are maxsum: read a sequence of n integer numbers (positive, zero,
Mixed integer programming emilie danna the increase in cache misses is a good candidate to explain the problems affects all
Extend properties of exponents to include all integer exponents b investigate and explain characteristics of exponential linear programming problems in two
Numerical results of problems using general purpose integer programming solver and the we explain these differences of the characteristics
Problems with time windows using mixed integer linear programming to solve a capacitated vehicle routing problem with explain the justifications of
Guidelines regarding assignments given in the programme explain the term class and object using programming problems is analyzed in terms of objects and the
Programming techniques and characteristics of good programs 2 solves practical problems using variable when an integer is used explain automatic
Aks/topics suggested time frame investigate and explain characteristics of exponential and ma2 a8 students will solve linear programming problems in two
11list the various methods are used to solve the assignment problems? explain how the profit maximization what do you mean by integer programming
According to the characteristics of teaching goals and teaching targets, transportation problems, integer programming, and make sure that the explain
Lagrangean relaxations for integer programming problems to explain how it can be applied, we then discuss these characteristics of the lagrangean function
Been applied to model the spread of health problems by epidemiologists [5], as we explain below, the following integer programming problem: max x;t e tt
Accelerated integrated geometry extend properties of exponents to include all integer investigate and explain characteristics of exponential
Integer programming • explain the new system and its relationship to characteristics of optimization problems • decisions
Optimum for one characteristic may not be optimum for other characteristics moreover then the non linear integer programming to explain the branching process
To round the particle position and pso can be used for integer programming integer in order to further explain integer optimization problems
Integer programming (a) for ip problems, the number of integer variables is generally state two characteristics of problems that would lead them to be
0 1 integer programming problems a number of issues that explain why no single improvement over an "optimal" integer programming solution based on a
Explain the concepts of c programming language see if there are any similarities between the current problem and other problems valued integer number
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A New Kind of Science
by Stephen Wolfram Publisher Comments
This long-awaited work from one of the world's most respected scientists presents a series of dramatic discoveries never before made public. Starting from a collection of simple computer experiments — illustrated in the book by striking computer... (read more)
The Drunkard's Walk: How Randomness Rules Our Lives
by Leonard Mlodinow Publisher Comments
In this irreverent and illuminating book, acclaimed writer and scientist Leonard Mlodinow shows us how randomness, change, and probability reveal a tremendous amount about our daily lives, and how we misunderstand the significance of everything from a... (read more)
Triangular Arrays with Applications
by Thomas Koshy Publisher Comments
Triangular arrays are a unifying thread throughout various areas of discrete mathematics such as number theory and combinatorics. They can be used to sharpen a variety of mathematical skills and tools, such as pattern recognition, conjecturing, proof... (read more)
Classical Topology & Combinatorial G 2ND Edition
by John Stillwell Publisher Comments
This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with... (read more)
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Algorithms (08 Edition)
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Papadimitrio is a one term algorithms text that takes an integrated approach and is priced well below any of the competitors within the market at $30.00.This product will take a more frequent revision cycle and will be the most current and up-to-date... (read more)
Topology 2ND Edition
by James Munkres Publisher Comments
This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness... (read more)
Numerical Analysis (2ND 88 Edition)
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If you want top grades and thorough understanding of numerical analysis, this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you accompanying related problems with fully worked solutions. You... (read more)
Principles of Mathematics Analysis (Cloth) (3RD 76 Edition)
by Walter Rudin Publisher Comments
The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field... (read more)
Algorithms
by Kenneth A. Berman Publisher Comments
Provides in-depth coverage of traditional and current topics in sequential algorithms, and also gives a solid introduction to the theory of parallel and distributed algorithms reflecting the emergence of modern computing environments such as parallel... (read more)
Introduction To Numerical Methods a Matlab 2ND Edition
by Ronald B. Guenther Publisher Comments
With thorough coverage and a direct approach, An Introduction to Numerical Methods: A MATLAB Approach, Second Edition introduces students to a wide range of useful and important algorithms. This second edition incorporates the use of MATLAB as
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Students explore the concept of piecewise functions. In this piecewise functions lesson plan, students discuss how to make a piecewise function continuous and differentiable. Students use their Ti-89 to find the limit of the function as it approaches a given x value. Students find the derivative of piecewise functions.
Learners explore the concept of piecewise functions. For this piecewise functions lesson, students graph piecewise functions by hand and find the domain and range. Learners make tables of values given a piecewise function. Students write piecewise functions given a graph.
Students explore the concept of piecewise functions. In this piecewise functions lesson, students find the derivatives of piecewise functions. Students determine points of discontinuity and jumps in the graph using their Ti-89 calculator.
Students explore the concept of piecewise functions. In this piecewise functions lesson plan, students write functions to represent the piecewise function graphs on their Ti-Nspire calculator. Students determine the formula given the piecewise function graph.
Students explore piecewise functions. In this Algebra II/Pre-calculus lesson, students write formulas for piecewise functions and check their work on the calculator. The lesson assumes that students have seen piecewise functions prior to this activity.
Calculus students find the limit of piecewise functions at a value. They find the limit of piecewise functions as x approaches a given value. They find the limit of linear, quadratic, exponential, and trigonometric piecewise functions.
A hands-on activity using the TI-CBR Motion Detector to provide information to graph and analyze. The class uses this information to calculate the slope of motion graphs and differentiate scalar and vector quantities. There is a real-world activity of a Roof Manufacturer's Test in regards to the pitch of roofs, as well as several other real-world scenarios.
Students investigate sonar technology. In this Algebra II lesson, students explore use sound waves to measure distance. The students conduct several experiments with a CBR 2 unit to collect data and graph distance vs. time. Students model the data with piecewise functions.
Graph piecewise functions as your learners work to identify the different values that will make a piecewise function a true statement. They identify function notations and graph basic polynomial functions. This lesson includes a series of critical thinking questions and vocabulary.
Students explore the concept of piecewise functions. In this piecewise functions lesson, students bring in their own electric bill and determine functions to represent their bill using their Ti-Nspire. Students graph piecewise functions that represent their energy bill. Students determine the cost of heating their home per square foot.
Eleventh graders explore the TI-InterAcitve!. In this Algebra II lesson, 11th graders examine features of the TI-InterActive! including drawing on a Graph, using Stat Plots, exploring the syntax for piecewise functions, and using sliders in order to obtain parametric variations. The lesson is designed to encourage students' creativity.
This pre-calculus worksheet is short, yet challenging. High schoolers calculate the limit of piecewise functions, rational functions, and graphs as x approaches a number from the positive or negative side. There are four questions.
Here is an activity that should catch the attention of your class! It focuses on the real-world problem of selecting the best cellular phone plan. This exercise would be especially good to use when introducing piecewise functions. Learners compare costs for various data plans, considering such features as unlimited talk and unlimited texts, to determine which plan is the most cost effective for different scenarios. The task requires giving graphical and numerical representations of the options and writing a justification for choosing a particular plan. The resource includes a detailed commentary for the teacher and three follow-up questions.
For this evaluating functions worksheet, 11th graders solve and complete 24 various types of problems. First, they evaluate the functions for the given values. Then, students graph each of piecewise functions. They also write an equation that relates the variables and graph the functions.
Learners hop online to complete a self-assessment. Using the interactive tool, they solve 16 multiple choice problems. They graph inequalities, find linear equations passing through two points, solve piecewise functions, find the domain and range of a relation, and find the first three terms in a sequence. There's even a hint provided for each problem.
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Featured Testimonial
Lesson Planet has been my place to go in order to find great additions to the curriculum. The lesson plans and worksheets I have found have been perfect for the lesson plans I had already planned. Because all lessons are reviewed by teachers, I know that every lesson I choose to look through will be pertinent and standard oriented without the fluff.
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focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
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Most Helpful Customer Reviews
By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.
If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.
I've taught an introductory differential geometry course from Lee's book, and in retrospect Do Carmo's "Riemannian Geometry" would have been a better choice. To be fair Lee does masterful job introducing basic concepts from curvature to Jacobi fields, but here are a few things I disliked. The book assumes working knowledge of smooth manifolds and Lie brackets, while many students need review of the former, and know nothing of the latter. Lee doesn't give enough examples beyond constant curvature spaces: there is virtually no mention of warped products, Riemannian submersions, Lie groups, or homogeneous spaces. Exercises are few, unmotivated, and their difficulty is in stark contrast with the easiness of the main text. I feel Do Carmo's book is superior in all respects, and last time I checked it was not much more expensive.
I used this book to teach about half a year of a graduate Riemannian manifolds course. It is a very good introductory text. I wish it has a bit more background on curves and surfaces, but otherwise it was excellent. It doesn't get into a lot of more advanced topics, but the treatment of Jacobi fields and so forth is really good.
I just got this fella, and I'm really just through the first four chaptors but so far I'm very pleased. He really tries to tie the definitions and theorems to something you can think about. He gives three "model spaces", the n-sphere, R^n, and hyperbolic space and keeps coming beck to them as he does new things. I like that after he defines connections he shows some in R^n. You know, things like that. Anyway, I'm not a specialist but this seems to me as good an introduction to Reimannian curvature as you could ask for. At least as good in my opinion as Del Carmo's book.
So thanks again Dr. Lee. You keep writing them and we'll keep reading them.
Prof. Lee sets the norm of mathematical exposition. I would give it 5 stars if it were more comprehensive. There is so much to say about Riemannian manifolds and it would be a pleasure to see them under the light the author sheds on such subtle concepts. One very nice feature of the book that underlies its structure is that it uses four theorems - pillars of Riemannian geometry as a guide of what should be included. This approach, besides improving considerably the organization of the book as compared to other books on the subject, it also motivates the reader who now has a target in his mind, namely the proofs of these important theorems. It is really nontrivial to introduce people to mathematical areas as broad as Riemannian geometry. Also, useful suggestions are given in the preface for further reading.
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Argo, IL PrecalculusThere is no universal agreement as to what constitutes discrete math. Discrete math is defined less by what topics are included than by what is excluded. Excluded are notions of continuity upon which calculus is built.
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College Mathematics for Business, Economics, Life Sciences and Social Sciences to be accessible, this book develops a thorough, functional understanding of mathematical concepts in preparation for their application in other areas. Coverage concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving.This book features a collection of important topics from mathematics of finance, linear algebra, linear programming, probability, and statistics, with an emphasis on cross-discipline principles and practices.For the professional who wants to acquire essential mathematical tools for application in business, economics, and the life and social sciences.
Preface
ix
Student and Instructor Resources
xv
Part One A Library of Elementary Functions
1
(128)
A Beginning Library of Elementary Functions
2
(76)
Functions
3
(18)
Elementary Functions: Graphs and Transformations
21
(14)
Linear Functions and Straight Lines
35
(18)
Quadratic Functions
53
(25)
Important Terms, Symbols, and Concepts
68
(2)
Review Exercise
70
(5)
Group Activity 1: Introduction to Regression Analysis
75
(1)
Group Activity 2: Mathematical Modeling in Business
76
(2)
Additional Elementary Functions
78
(51)
Polynomial and Rational Functions
79
(16)
Exponential Functions
95
(14)
Logarithmic Functions
109
(20)
Important Terms, Symbols, and Concepts
122
(1)
Review Exercise
123
(3)
Group Activity 1: Comparing the Growth of Exponential and Polynomial Functions, and Logarithmic and Root Functions
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The Centre supports students who do not have the background in mathematics necessary for their Introductory Biostatistics unit of study. Staff at the Centre understand the objectives and structure of the course, and have organised their support to complement the lectures and tutorials. Students from the School of Public Health who have used the Centre in the past have been very happy with the services. There are two main ways in which the Centre can provide support:
The Centre has organised regular weekly sessions when their staff can help to sort out any problems you are having with concepts covered in the course. The sessions are on Thursdays @ 10:00 – 11:00 OR 16:00 – 17:00 starting in Week 3. Further details can be obtained from the Centre.
The Centre has handbooks that students can buy or download for free from the MLC website.
Before using either of these services, students should approach Jackie Nicholas, who is the Head of the Centre, to discuss their needs. Jackie will introduce herself, and the MLC, at the first lecture. She can be contacted on 02 9351-4061. Students expecting difficulties should see Jackie as soon as possible. There are no costs involved in using the Centre. You will have to pay if you want any of their handbooks.
Please do not ask the staff at the Centre for assistance with your assessed work.
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Teaching Mathematics in Grades 9-12
Math 3330-001
TTR 9:40-11:05
Spring 2006
Instructor: Dr. Jeremy Winters
Office: KOM 226B
E-mail: [email protected]
Phone: 898-2491
Office Hours: TTR 11:30-3:00 or by appointment
Due to supervising student teachers, I will be out at the
various schools on occasion. Therefore the I may be out of
the office during the above times, please feel free to make
an appointment to guarantee that I will be in the office, or
drop by and try and catch me in my office at any time.
Textbook: There is no textbook for this course. A number of readings
and handouts will be provided for you to include in a 3-ring
binder. Other items will be located on the internet or in the
library. Since you are not purchasing a book for this course,
it is expected that you will obtain a membership in NCTM and
(MT)2. Proof of these memberships will be a requirement
described below.
Other items: (a) You will need access to the Principles and Standards for
School Mathematics (NCTM, 2000).
(b) Access to MTSU e-mail.and the internet
Prerequisites:
Admission to teacher education, completion of the mathematics core (MATH 1910,
1920, 3110, 2010, 3460; 2050), and MATH 3320. (Three credits)
Special Note:
If you have a disability that may require assistance or accommodation,
or you have questions related to accommodations for testing, note-takers, readers, etc.,
please contact me as soon as possible. You may also contact the Office of Disabled
Student Services (898-2783) with questions about such services. You must be
registered with Disability Services to receive special accommodations.
Course Objectives:
Students in this course will:
1. investigate various methods of teaching mathematics.
2. read, analyze, and apply the NCTM Principles and Standards for School
Mathematics, as well as other professional readings..
3. differentiate between conceptual and procedural types of mathematical
knowledge. Students will make attempts to implement strategies and methods in
tasks and lessons that promote the development of conceptual knowledge.
4. investigate the use of various types of manipulatives and technology into the
teaching of mathematics and appropriately include them in instructional activities
and tasks.
5. examine various types of questioning techniques and exploratory activities.
6. become familiar with different methods of assessment and what can be gained
from using various forms of assessment in the classroom.
7. investigate various curricula and how to decide what and when to teach particular
mathematical topics.
8. address issues related to classroom management, discipline, gender,
multiculturalism, and special needs. Students will use this information to help
them in designing appropriate lesson plans and activities for classroom
instruction.
9. have opportunities to reflect, communicate, and record their experiences, which
in turn may help provide them with valuable insight to continue this practice
throughout their careers (in class as well as out in the field).
10. become experienced in working individually as well as collaboratively.
Attendance:
On-time attendance is expected. Three (3) or more absences and excessive tardies or
early departures will result in the lowering of one's grade by at least one letter grade.
Absolutely no late work will be accepted. Tests, with a valid university excuse, will be
able to be made up in a timely manner by appointment with the instructor. If you are
absent and an assignment is due, you may e-mail me the assignment prior to class. I
will not accept e-mailed assignments after the start time of class. If classwork is
assigned for the next class meeting it is the responsibility of the student to contact the
professor or another student in order to be prepared for the next class meeting.
Course Evaluations
1. Membership and Participation (5%)
Students must join and participate in the Preservice Mathematics Teachers
Organization (MT)^3. Moreover, each student is required to obtain a student
membership to NCTM. This requirement may also include assisting in the TMTA
high school math contest, a TI workshop, or attending another professional
meeting.
2. SRQ's (15%)
Summarize, Reflect, and Questions
Students will be asked to SRQ various readings. Readings will consist of whole
class readings and individual readings. Whole class readings will be a common
article or chapter that all students will read. Individual readins will be selected by
each individual. For these readings, students will be asked to share the SRQ with
the class and lead a discussion over the reading.
3. Teacher Observations (20%)
Students are required to do two types of observations. First, students are required
to watch selected on-line videos of mathematics teachers and answer the
questions which accompany the videos. Second, students are required to log ten
(10) hours observing a high school mathematics teacher. Details concerning these
ten (10) hours will given on a separate sheet of paper. Students are allowed to
choose any high school math teacher with consent from the professor. If you do
not know or have a particular high school math teacher in mind, I will aide in
helping you to find one.
4. Unit Plan (20%)
Students will design and organize a detailed unit plan for a secondary mathematics
class. Details concerning the Unit Plan will be given on a separate sheet of paper.
5. Portfolio (15%)
Students will be required to turn in a portfolio. Details concerning the portfolio will
be given on a separate sheet.
6. Presentations (15%)
Students will do three presentations during this course.
(a) Activity/Investigation
(b) Short Lesson (Individual)
(c) Short Lesson (Group)
7. Final Exam (10%)
The exam will be similar to the Praxis Mathematics Pedagogy Test required for
Teacher Licensure.
Gradomg Scale:
A: 100-89.5 C+: 79.45-76.5
B+: 89.49-86.5 C: 76.49-73.5
B: 86.49-83.5 C-: 73.49-69.5
B-: 83.49-79.5 D: 69.49-59.5
F: Below 59.49
(I will not round, all calculated grades past the hundredths place will be dropped
not rounded!)
Important Dates:
January 16 Last day to DROP classes for a 100% refund
January 16 Last day to register without paying $100 late fee
Last day to WITHDRAW FROM ALL CLASSES for 100%
January 16
refund
January 18 Last day to REGISTER (late fee charged)
January 20 Last day to ADD classes
January 28 (MT)^2 Annual Meeting at Pope John Paul II School
January 30 Last day for a 75% refund
January 30 Last day to drop or withdraw WITHOUT a grade
February
Last day for a 25% refund
12
February Classroom Management Seminar, STEA Meeting
16 5:00-7:00 in KUC
March 6 Last day to drop or withdraw with a grade of "W"
March 6-11 Spring Break
April 6 Middle School Math Contest
April 18 TMTA High School Math Contest
April 19 Last day to withdraw from the University
April 26 Last day of classes
April 27 Study Day
April 26-29 NCTM Annual Meeting in St. Louis MO
Tuesday,
Final Exam from 10:30-12:30 in KOM 200
May 2
Academic Integrity:
According to the Rights and Responsibility section of the Students Handbook, cheating
is defined as intentionally using or attempting to use unauthorized materials,
information, or study aids in any academic exercise. The term academic exercise
includes all forms of work submitted for credit or hours. If a student is believed to be in
violation of MTSU's policy on academic misconduct, procedures will be following as
outlined in the Students Handbook.
Academic Misconduct:
The instructor has the primary responsibility for control over the classroom behavior and
can direct the temporary removal or exclusion from the classroom of any student
engaged in disruptive conduct or conduct which otherwise violates the general rules and
regulations of the institution. The instructor may report such misconduct to the assistant
dean for Judicial Affairs for implementation of such disciplinary sanctions as may be
appropriate, including extended or permanent exclusion from the classroom
| 677.169 | 1 |
An understanding of the principle elements of algebra is essential to upper-level math and good standardized test scores. Introduce your junior high students to advanced math with this kit's 160 colorful lessons. The colorful student workbook reviews basic math skills before introducing algebra, geometry, and trigonometry concepts like absolute value, transformations and nets, compound interest, permutations, combinations, two variable equations, volume and surface area of solids, four operations with monomials and polynomials, representations of data, trigonometric ratios and more. Grade 7.
The student workbook includes a set of college test prep questions that follows each block of ten lessons; a new collection of math-minute interviews help students understand how ordinary people use pre-algebra concepts in their work. 358 pages, softcover.
The teacher's guide includes the main concepts, lesson objectives, materials needed, teaching tips, the assignment for the day, and the reduced student pages with the correct answers supplied. Each lesson will take approximately 45-60 minutes, and is designed to be teacher-directed. 400 pages, softcover.
Best Math Curriculum
Date:September 23, 2013
strickland99
Location:Leicester, NC
Age:35-44
Gender:female
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
We've used Horizon's Math for five years and it is, by far, the best math curriculum available. I have two girls that can sometimes get pretty frustrated with the more difficult lessons, but with Horizon's that frustration is usually short-lived. It will introduce a difficult activity and then go back to easier tasks that build back up to the more difficult ones. It's bright and colorful pages keep their interest piqued. Since we started Horizon's, my girls stay 2-3 grade levels ahead in math!!
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+2points
2of2voted this as helpful.
Review 2 for Horizons Pre-Algebra Complete Set
Overall Rating:
4out of5
For the solid math student
Date:April 27, 2013
Jamila T
Location:Charlotte, NC
Age:35-44
Gender:female
Quality:
4out of5
Value:
4out of5
Meets Expectations:
5out of5
I have used Horizons K-8. It has always been a bit fater paced than the "average" math curriculum in ALL their subjects. I would not recomend this for any student who has not completed Horizons grade 7 or is a strong math student. It moves just as quickly as any other Horizons. I found the classwork to be far tougher than the tests. I appreciated the practical application aspects. It does well in covering the topics and teaching application. There is not a ton of practice on each concept, but more than enough for a GOOD math student that would get bored the extreme repetition. I am more than happy with it. I have bought Grade 8 and can't wait for grade 9.
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+3points
3of3voted this as helpful.
Review 3 for Horizons Pre-Algebra Complete Set
Overall Rating:
2out of5
disappointed
Date:January 28, 2013
MDV
Age:35-44
Gender:female
I have used Horizons for K-6th and was so excited to see they added Pre-Algebra. However, it has not worked out well at all for us. It is very fast paced but gives very little practice of new concepts. We also have found that at times a new concept is just introduced and in the first few problems they give the student, there is an exception to the rule or a very complex problem. The student does not even get a chance to learn the basics before complexities are given to them. My son who loves math and has alway done very well now is very discouraged and hates math. I have stopped the program half way through the year and will not continue with Horizons after 6th grade. Hopefully they can improve this product in the future to be up to par with their K-6th math.
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+3points
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Review 4 for Horizons Pre-Algebra Complete Set
Overall Rating:
4out of5
Overall a great program
Date:October 11, 2012
Shelley
Age:45-54
Gender:female
Quality:
4out of5
Value:
4out of5
Meets Expectations:
5out of5
I have used Horizons for K-2 and 5 and always felt it was a complete and advanced program. After competing Horizons 2, my older daughter was a year ahead in math when she entered school. With my himeschooled younger child, we switched to Saxon 7/6 for 5th grade and my daughter really balked at the boring black and white layout and repetitiveness. I think Saxon is a great program, but my daughter really likes color. It seems crazy to switch for this reason, but if the layout is unappealing, it can be very difficult to persevere. When I saw Horizon had Pre-Algebra, I jumped on it. We have been doing it for twenty days and overall we are happy. I would say that it moves quite quickly and does not offer as much practice as Saxon. It also does not offer the complete explanations in the Teachers Guide offered by Saxon, but it does cover all the same material. Most importantly, my daughter is working through it happily and is doing well on the tests. I always work through the problems first to be sure I can explain them well. The TM could definitely offer more detail. If you are not a math person, this curriculum may not be for you.
We love the stories of real life people who use math in their work. The word problems are engaging, although there could be more. I do think this curriculum will well prepare a student for Algebra. I have been comparing the Scope and Sequence to a public school math book and it definitely lines up. I love that we can reasonably complete one lesson per day. I will definitely consider horizons Algebra I next year.
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Review 5 for Horizons Pre-Algebra Complete Set
Overall Rating:
3out of5
disappointed
Date:May 23, 2012
docmom
Quality:
3out of5
Value:
3out of5
Meets Expectations:
2out of5
I used Horizons for 3d, 4th, 5th, and 6th with no complaints. But, this book was a disappointment. I really agonized whether to switch to Saxon for pre-algebra but went with Horizons b/c it has color (lol) and my daughter loves that. The pre-algebra book is very fast paced (there is a new topic every day), but there are not enough practice problems and worksheets. I think they tried to cram too much into this book. We're trying a "real" text for algebra next year (rather than a homeschool text).
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-3points
1of5voted this as helpful.
Review 6 for Horizons Pre-Algebra Complete Set
Overall Rating:
5out of5
Excellent Pre Algebra program
Date:May 4, 2012
TracyR
Location:Corry'll admit that Horizons was the last on my list for math programs with my children. Happy to say though that I went ahead and took a leap with buying an Alpha Omega product and couldn't be happier. We used a different math program with my oldest daughter and she just was not 'getting it' no matter what I did with her. I tried many things to make the curriculum work for us and it just wasn't working. To find out she needed color to concentrate on the math problems. I searched and Horizon's Pre Algebra was just what we needed. To focus on one concept at a time and to review concepts that were learned earlier so that they stayed fresh in her mind.
I will say you need the whole set for the program to work. The teacher manual contains all the problems and solutions , the resource book contains all the tests, quizes and final exams, and practice worksheets. The teacher manual though does not explain things the way I wished it did because I'm not very math minded, but I have found with the advent of websites like Khan Academy and Hippo Campus really the manual is obsolete when it comes to explaining things anymore.
What I like about this course is not only does it teach Pre Algebra but it also introduces the beginning of Trigonometry as well. I also like the College Prep tests they include in the workbook for the children to practice with. It definitely gives them an idea of what to expect for college entry tests some day. I also like the continuous review so that she does not forget what she's learned. My daughter says she likes the color to help her focus on the math problems, and the fact it focuses on one math concept at a time.
Everything we have is of high quality and is an excellent product. The only negative is that Horizons only goes to Algebra 1 as of right now (2012). I hope they come out with more levels in the future.
I do plan on giving it a try with my other three children for the upcoming school year.
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Review 7 for Horizons Pre-Algebra Complete Set
Overall Rating:
1out of5
Lessons are not leveled and minimal teacher help
Date:September 27, 2011
Marie
Location:GA
Age:45-54
Gender:female
Quality:
2out of5
Value:
1out of5
Meets Expectations:
1out of5
We are on lesson 50 of this book and I am very unhappy. The lessons on fractions, decimals and percents are on the easy side but then they introduce solving square roots and scientific notation which is very difficult. The word problems start out okay but then get very difficult. There also isn't near enough word probloms but that is true with most math books. Some lessons introduce a concept and then give no practice exercises. The teacher's manual is worthless; the problems are not worked out and the print is tiny. I have homeschooled for many years and have used many math programs; this was a waste of time and money. I rarely write reviews but felt compelled to in this case.
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+2points
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Review 8 for Horizons Pre-Algebra Complete Set
Overall Rating:
5out of5
looks awesome
Date:February 15, 2011
mcat
Location:Atlanta,GA
Age:35-44
Gender:female
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
I just received the set and after using Horizons for 3 years now this set looks even better than the other levels. I just love how well structured the pages are, also great for older kids (the K-6 have more pictures, this has real people like missionaries, etc telling, where they need math, which I find very encouraging!) Also the teacher book now has a printed student page in small (like BJU does) so you´ll find the correct answers right within the lesson . I like that the extra worksheets are not in the teacher book anymore, but in the seperate test and resource book, so you can just get this book new for each child. I will update my comment as soon as we´ve worked with this books for a while, happy home schooling!!
| 677.169 | 1 |
Performs useful calculations such as finding the Area, Common Factors of a set of numbers, Distance between two points, Quadratic Roots, Midpoint, Perimeter, Find all the primes up to and including a number, the slope of a line, and the volume of a cube, cone, cylinder, and sphere.Finds the area of a Rectangle, Circle, Triangle, Oval, Cylinder, Cone, and Pyramid.Formula for each function is displayed at the top of the screen for easy reference. Easy to use, convenient and powerful math program. Great for math class and home work!An easy and convenient handwriting recognition system allows numbers to be entered by just writing on the screen with your finger
| 677.169 | 1 |
PrefaceThe International Mathematical Olympiad (IMO) is nearing its fiftieth an-niversary and has already created a very rich legacy and firmly establisheditself as the most prestigious mathematical competition in which a high-schoolstudent could aspire to participate. Apart from the opportunity to tackle in-teresting and very challenging mathematical problems, the IMO representsa great opportunity for high-school students to see how they measure upagainst students from the rest of the world. Perhaps even more importantly,it is an opportunity to make friends and socialize with students who havesimilar interests, possibly even to become acquainted with their future col-leagues on this first leg of their journey into the world of professional andscientific mathematics. Above all, however pleasing or disappointing the finalscore may be, preparing for an IMO and participating in one is an adventurethat will undoubtedly linger in one's memory for the rest of one's life. It isto the high-school-aged aspiring mathematician and IMO participant that wedevote this entire book. The goal of this book is to include all problems ever shortlisted for theIMOs in a single volume. Up to this point, only scattered manuscripts tradedamong different teams have been available, and a number of manuscripts werelost for many years or unavailable to many. In this book, all manuscripts have been collected into a single compendiumof mathematics problems of the kind that usually appear on the IMOs. There-fore, we believe that this book will be the definitive and authoritative sourcefor high-school students preparing for the IMO, and we suspect that it will beof particular benefit in countries lacking adequate preparation literature. Ahigh-school student could spend an enjoyable year going through the numer-ous problems and novel ideas presented in the solutions and emerge ready totackle even the most difficult problems on an IMO. In addition, the skill ac-quired in the process of successfully attacking difficult mathematics problemswill prove to be invaluable in a serious and prosperous career in mathematics. However, we must caution our aspiring IMO participant on the use of thisbook. Any book of problems, no matter how large, quickly depletes itself if
VI Prefacethe reader merely glances at a problem and then five minutes later, havingdetermined that the problem seems unsolvable, glances at the solution. The authors therefore propose the following plan for working through thebook. Each problem is to be attempted at least half an hour before the readerlooks at the solution. The reader is strongly encouraged to keep trying to solvethe problem without looking at the solution as long as he or she is coming upwith fresh ideas and possibilities for solving the problem. Only after all venuesseem to have been exhausted is the reader to look at the solution, and thenonly in order to study it in close detail, carefully noting any previously unseenideas or methods used. To condense the subject matter of this already verylarge book, most solutions have been streamlined, omitting obvious derivationsand algebraic manipulations. Thus, reading the solutions requires a certainmathematical maturity, and in any case, the solutions, especially in geometry,are intended to be followed through with pencil and paper, the reader fillingin all the omitted details. We highly recommend that the reader mark suchunsolved problems and return to them in a few months to see whether theycan be solved this time without looking at the solutions. We believe this tobe the most efficient and systematic way (as with any book of problems) toraise one's level of skill and mathematical maturity. We now leave our reader with final words of encouragement to persist inthis journey even when the difficulties seem insurmountable and a sincere wishto the reader for all mathematical success one can hope to aspire to.Belgrade, Duˇan Djuki´ s cOctober 2004 Vladimir Jankovi´c Ivan Mati´c Nikola Petrovi´ c For the most current information regarding The IMO Compendium youare invited to go to our website: At this site you can alsofind, for several of the years, scanned versions of available original shortlistand longlist problems, which should give an illustration of the original statethe IMO materials we used were in. We are aware that this book may still contain errors. If you find any, pleasenotify us at [email protected]. A full list of discovered errors can be foundat our website. If you have any questions, comments, or suggestions regardingboth our book and our website, please do not hesitate to write to us at theabove email address. We would be more than happy to hear from you.
Preface VIIAcknowledgementsThe making of this book would have never been possible without the help ofnumerous individuals, whom we wish to thank. First and foremost, obtaining manuscripts containing suggestions for IMOswas vital in order for us to provide the most complete listing of problemspossible. We obtained manuscripts for many of the years from the formerand current IMO team leaders of Yugoslavia / Serbia and Montenegro, whocarefully preserved these valuable papers throughout the years. Special thanksare due to Prof. Vladimir Mi´i´, for some of the oldest manuscripts, and ccto Prof. Zoran Kadelburg. We also thank Prof. Djordje Dugoˇija and Prof. sPavle Mladenovi´. In collecting shortlisted and longlisted problems we were calso assisted by Prof. Ioan Tomescu from Romania and Hà Duy Hưng fromVietnam. A lot of work was invested in cleaning up our giant manuscript of errors.Special thanks in this respect go to David Kramer, our copy-editor, and toProf. Titu Andreescu and his group for checking, in great detail, the validityof the solutions in this manuscript, and for their proposed corrections andalternative solutions to several problems. We also thank Prof. AbderrahimOuardini from France for sending us the list of countries of origin for theshortlisted problems of 1998, Prof. Dorin Andrica for helping us compile the ˇ clist of books for reference, and Prof. Ljubomir Cuki´ for proofreading part ofthe manuscript and helping us correct several errors. We would also like to express our thanks to all anonymous authors of theIMO problems. It is a pity that authors' names are not registered togetherwith their proposed problems. Without them, the IMO would obviously notbe what it is today. In many cases, the original solutions of the authors wereused, and we duly acknowledge this immense contribution to our book, thoughonce again, we regret that we cannot do this individually. In the same vein,we also thank all the students participating in the IMOs, since we have alsoincluded some of their original solutions in this book. The illustrations of geometry problems were done in WinGCLC, a programcreated by Prof. Predrag Janiˇi´. This program is specifically designed for cccreating geometric pictures of unparalleled complexity quickly and efficiently.Even though it is still in its testing phase, its capabilities and utility arealready remarkable and worthy of highest compliment. Finally, we would like to thank our families for all their love and supportduring the making of this book.
1Introduction1.1 The International Mathematical OlympiadThe International Mathematical Olympiad (IMO) is the most important andprestigious mathematical competition for high-school students. It has played asignificant role in generating wide interest in mathematics among high schoolstudents, as well as identifying talent. In the beginning, the IMO was a much smaller competition than it is today.In 1959, the following seven countries gathered to compete in the first IMO:Bulgaria, Czechoslovakia, German Democratic Republic, Hungary, Poland,Romania, and the Soviet Union. Since then, the competition has been heldannually. Gradually, other Eastern-block countries, countries from WesternEurope, and ultimately numerous countries from around the world and everycontinent joined in. (The only year in which the IMO was not held was 1980,when for financial reasons no one stepped in to host it. Today this is hardly aproblem, and hosts are lined up several years in advance.) In the 45th IMO,held in Athens, no fewer than 85 countries took part. The format of the competition quickly became stable and unchanging.Each country may send up to six contestants and each contestant competesindividually (without any help or collaboration). The country also sends ateam leader, who participates in problem selection and is thus isolated fromthe rest of the team until the end of the competition, and a deputy leader,who looks after the contestants. The IMO competition lasts two days. On each day students are givenfour and a half hours to solve three problems, for a total of six problems.The first problem is usually the easiest on each day and the last problemthe hardest, though there have been many notable exceptions. ((IMO96-5) isone of the most difficult problems from all the Olympiads, having been fullysolved by only six students out of several hundred!) Each problem is worth 7points, making 42 points the maximum possible score. The number of pointsobtained by a contestant on each problem is the result of intense negotiationsand, ultimately, agreement among the problem coordinators, assigned by the
2 1 Introductionhost country, and the team leader and deputy, who defend the interests of theircontestants. This system ensures a relatively objective grade that is seldomoff by more than two or three points. Though countries naturally compare each other's scores, only individualprizes, namely medals and honorable mentions, are awarded on the IMO.Fewer than one twelfth of participants are awarded the gold medal, fewerthan one fourth are awarded the gold or silver medal, and fewer than one halfare awarded the gold, silver or bronze medal. Among the students not awardeda medal, those who score 7 points on at least one problem are awarded anhonorable mention. This system of determining awards works rather well. Itensures, on the one hand, strict criteria and appropriate recognition for eachlevel of performance, giving every contestant something to strive for. On theother hand, it also ensures a good degree of generosity that does not greatlydepend on the variable difficulty of the problems proposed. According to the statistics, the hardest Olympiad was that in 1971, fol-lowed by those in 1996, 1993, and 1999. The Olympiad in which the winningteam received the lowest score was that in 1977, followed by those in 1960 and1999. The selection of the problems consists of several steps. Participant coun-tries send their proposals, which are supposed to be novel, to the IMO orga-nizers. The organizing country does not propose problems. From the receivedproposals (the longlisted problems), the problem committee selects a shorterlist (the shortlisted problems), which is presented to the IMO jury, consistingof all the team leaders. From the short-listed problems the jury chooses sixproblems for the IMO. Apart from its mathematical and competitive side, the IMO is also a verylarge social event. After their work is done, the students have three daysto enjoy events and excursions organized by the host country, as well as tointeract and socialize with IMO participants from around the world. All thismakes for a truly memorable experience.1.2 The IMO CompendiumOlympiad problems have been published in many books [65]. However, theremaining shortlisted and longlisted problems have not been systematicallycollected and published, and therefore many of them are unknown to math-ematicians interested in this subject. Some partial collections of shortlistedand longlisted problems can be found in the references, though usually onlyfor one year. References [1], [30], [41], [60] contain problems from multipleyears. In total, these books cover roughly 50% of the problems found in thisbook. The goal of this book is to present, in a single volume, our comprehen-sive collection of problems proposed for the IMO. It consists of all problemsselected for the IMO competitions, shortlisted problems from the 10th IMO
1.2 The IMO Compendium 3and from the 12th through 44th IMOs, and longlisted problems from nineteenIMOs. We do not have shortlisted problems from the 9th and the 11th IMOs,and we could not discover whether competition problems at those two IMOswere selected from the longlisted problems or whether there existed shortlistedproblems that have not been preserved. Since IMO organizers usually do notdistribute longlisted problems to the representatives of participant countries,our collection is incomplete. The practice of distributing these longlists effec-tively ended in 1989. A selection of problems from the first eight IMOs hasbeen taken from [60]. The book is organized as follows. For each year, the problems that weregiven on the IMO contest are presented, along with the longlisted and/orshortlisted problems, if applicable. We present solutions to all shortlistedproblems. The problems appearing on the IMOs are solved among the othershortlisted problems. The longlisted problems have not been provided withsolutions, except for the two IMOs held in Yugoslavia (for patriotic reasons),since that would have made the book unreasonably long. This book has thusthe added benefit for professors and team coaches of being a suitable bookfrom which to assign problems. For each problem, we indicate the countrythat proposed it with a three-letter code. A complete list of country codesand the corresponding countries is given in the appendix. In all shortlists, wealso indicate which problems were selected for the contest. We occasionallymake references in our solutions to other problems in a straightforward way.After indicating with LL, SL, or IMO whether the problem is from a longlist,shortlist, or contest, we indicate the year of the IMO and then the numberof the problem. For example, (SL89-15) refers to the fifteenth problem of theshortlist of 1989. We also present a rough list of all formulas and theorems not obviouslyderivable that were called upon in our proofs. Since we were largely concernedwith only the theorems used in proving the problems of this book, we believethat the list is a good compilation of the most useful theorems for IMO prob-lem solving. The gathering of such a large collection of problems into a book requireda massive amount of editing. We reformulated the problems whose originalformulations were not precise or clear. We translated the problems that werenot in English. Some of the solutions are taken from the author of the problemor other sources, while others are original solutions of the authors of thisbook. Many of the non-original solutions were significantly edited before beingincluded. We do not make any guarantee that the problems in this bookfully correspond to the actual shortlisted or longlisted problems. However, webelieve this book to be the closest possible approximation to such a list.
10 2 Basic Concepts and FactsDefinition 2.29. A group G = (G, ∗) is commutative or abelian if a ∗ b = b ∗ afor all a, b ∈ G.Definition 2.30. A set A generates a group (G, ∗) if every element of G canbe obtained using powers of the elements of A and the operation ∗. In otherwords, if A is the generator of a group G then every element g ∈ G can bewritten as ai1 ∗ · · · ∗ ain , where aj ∈ A and ij ∈ Z for every j = 1, 2, . . . , n. 1 nDefinition 2.31. The order of a ∈ G is the smallest n ∈ N such that an = e,if it exists. The order of a group is the number of its elements, if it is finite.Each element of a finite group has a finite order.Theorem 2.32 (Lagrange's theorem). In a finite group, the order of anelement divides the order of the group.Definition 2.33. A ring is a nonempty set R equipped with two operations+ and · such that (R, +) is an abelian group and for any a, b, c ∈ R,(i) (a · b) · c = a · (b · c);(ii) (a + b) · c = a · c + b · c and c · (a + b) = c · a + c · b.A ring is commutative if a · b = b · a for any a, b ∈ R and with identity if thereexists a multiplicative identity i ∈ R such that i · a = a · i = a for all a ∈ R.Definition 2.34. A field is a commutative ring with identity in which everyelement a other than the additive identity has a multiplicative inverse a−1such that a · a−1 = a−1 · a = i.Theorem 2.35. The following are common examples of groups, rings, andfields: Groups: (Zn , +), (Zp {0}, ·), (Q, +), (R, +), (R {0}, ·). Rings: (Zn , +, ·), (Z, +, ·), (Z[x], +, ·), (R[x], +, ·). √ Fields: (Zp , +, ·), (Q, +, ·), (Q( 2), +, ·), (R, +, ·), (C, +, ·).2.2 AnalysisDefinition 2.36. A sequence {an }∞ has a limit a = limn→∞ an (also de- n=1noted by an → a) if (∀ε > 0)(∃nε ∈ N)(∀n ≥ nε ) |an − a| < ε.A function f : (a, b) → R has a limit y = limx→c f (x) if (∀ε > 0)(∃δ > 0)(∀x ∈ (a, b)) 0 < |x − c| < δ ⇒ |f (x) − y| < ε.
12 2 Basic Concepts and Factsthat a point a in D is an extremum of the function f within the set of pointsin D such that f1 = f2 = · · · = fn = 0. Then there exist real numbersλ1 , . . . , λk (so-called Lagrange multipliers) such that a is a stationary point ofthe function F = f + λ1 f1 + · · · + λk fk , i.e., such that all partial derivativesof F at a are zero.Definition 2.48. Let f be a real function defined on [a, b] and let a = x0 ≤ nx1 ≤ · · · ≤ xn = b and ξk ∈ [xk−1 , xk ]. The sum S = k=1 (xk − xk−1 )f (ξk )is called a Darboux sum. If I = limδ→0 S exists (where δ = maxk (xk − xk−1 )),we say that f is integrable and I its integral. Every continuous function isintegrable on a finite interval.2.3 Geometry2.3.1 Triangle GeometryDefinition 2.49. The orthocenter of a triangle is the common point of itsthree altitudes.Definition 2.50. The circumcenter of a triangle is the center of its circum-scribed circle (i.e. circumcircle). It is the common point of the perpendicularbisectors of the sides of the triangle.Definition 2.51. The incenter of a triangle is the center of its inscribed circle(i.e. incircle). It is the common point of the internal bisectors of its angles.Definition 2.52. The centroid of a triangle (median point) is the commonpoint of its medians.Theorem 2.53. The orthocenter, circumcenter, incenter and centroid arewell-defined (and unique) for every non-degenerate triangle.Theorem 2.54 (Euler's line). The orthocenter H, centroid G, and cir-cumcircle O of an arbitrary triangle lie on a line (Euler's line) and satisfy−→ − − −→HG = 2GO.Theorem 2.55 (The nine-point circle). The feet of the altitudes fromA, B, C and the midpoints of AB, BC, CA, AH, BH, CH lie on a circle(The nine-point circle).Theorem 2.56 (Feuerbach's theorem). The nine-point circle of a triangleis tangent to the incircle and all three excircles of the triangle.Theorem 2.57. Given a triangle ABC, let ABC , AB C, and A BCbe equilateral triangles constructed outwards. Then AA , BB , CC intersectin one point, called Torricelli's point.
2.3 Geometry 15Theorem 2.72 (Casey's theorem). Let k1 , k2 , k3 , k4 be four circles that alltouch a given circle k. Let tij be the length of a segment determined by anexternal common tangent of circles ki and kj (i, j ∈ {1, 2, 3, 4}) if both ki andkj touch k internally, or both touch k externally. Otherwise, tij is set to be theinternal common tangent. Then one of the products t12 t34 , t13 t24 , and t14 t23is the sum of the other two. Some of the circles k1 , k2 , k3 , k4 may be degenerate, i.e. of 0 radius andthus reduced to being points. In particular, for three points A, B, C on a circlek and a circle k touching k at a point on the arc of AC not containing B, wehave AC · b = AB · c + a · BC, where a, b, and c are the lengths of the tangentsegments from points A, B, and C to k . Ptolemy's theorem is a special caseof Casey's theorem when all four circles are degenerate.Theorem 2.73. A convex quadrilateral ABCD is tangent (i.e., there existsan incircle of ABCD) if and only if AB + CD = BC + DA.Theorem 2.74. For arbitrary points A, B, C, D in space, AC ⊥ BD if andonly if AB 2 + CD2 = BC 2 + DA2 .Theorem 2.75 (Newton's theorem). Let ABCD be a quadrilateral, AD ∩BC = E, and AB ∩ DC = F (such points A, B, C, D, E, F form a com-plete quadrilateral). Then the midpoints of AC, BD, and EF are collinear.If ABCD is tangent, then the incenter also lies on this line.Theorem 2.76 (Brocard's theorem). Let ABCD be a quadrilateral in-scribed in a circle with center O, and let P = AB ∩ CD, Q = AD ∩ BC,R = AC ∩ BD. Then O is the orthocenter of P QR.2.3.5 Circle GeometryTheorem 2.77 (Pascal's theorem). If A1 , A2 , A3 , B1 , B2 , B3 are distinctpoints on a conic γ (e.g., circle), then points X1 = A2 B3 ∩ A3 B2 , X2 =A1 B3 ∩ A3 B1 , and X3 = A1 B2 ∩ A2 B1 are collinear. The special result whenγ consists of two lines is called Pappus's theorem.Theorem 2.78 (Brianchon's theorem). Let ABCDEF be an arbitraryconvex hexagon circumscribed about a conic (e.g., circle). Then AD, BE andCF meet in a point.Theorem 2.79 (The butterfly theorem). Let AB be a segment of circlek and C its midpoint. Let p and q be two different lines through C that,respectively, intersect k on one side of AB in P and Q and on the other in Pand Q . Let E and F respectively be the intersections of P Q and P Q withAB. Then it follows that CE = CF .
16 2 Basic Concepts and FactsDefinition 2.80. The power of a point X with respect to a circle k(O, r) isdefined by P(X) = OX 2 −r2 . For an arbitrary line l through X that intersects −→ −→ − −k at A and B (A = B when l is a tangent), it follows that P(X) = XA · XB.Definition 2.81. The radical axis of two circles is the locus of points thathave equal powers with respect to both circles. The radical axis of circlesk1 (O1 , r1 ) and k2 (O2 , r2 ) is a line perpendicular to O1 O2 . The radical axesof three distinct circles are concurrent or mutually parallel. If concurrent, theintersection of the three axes is called the radical center.Definition 2.82. The pole of a line l O with respect to a circle k(O, r) is apoint A on the other side of l from O such that OA ⊥ l and d(O, l) · OA = r2 .In particular, if l intersects k in two points, its pole will be the intersection ofthe tangents to k at these two points.Definition 2.83. The polar of the point A from the previous definition is theline l. In particular, if A is a point outside k and AM , AN are tangents to k(M, N ∈ k), then M N is the polar of A.Poles and polares are generally defined in a similar way with respect to arbi-trary non-degenerate conics.Theorem 2.84. If A belongs to a polar of B, then B belongs to a polar of A.2.3.6 InversionDefinition 2.85. An inversion of the plane π around the circle k(O, r) (whichbelongs to π), is a transformation of the set π{O} onto itself such that everypoint P is transformed into a point P on (OP such that OP · OP = r2 . Inthe following statements we implicitly assume exclusion of O.Theorem 2.86. The fixed points of the inversion are on the circle k. Theinside of k is transformed into the outside and vice versa.Theorem 2.87. If A, B transform into A , B after an inversion, then ∠OAB= ∠OB A , and also ABB A is cyclic and perpendicular to k. A circle per-pendicular to k transforms into itself. Inversion preserves angles between con-tinuous curves (which includes lines and circles).Theorem 2.88. An inversion transforms lines not containing O into circlescontaining O, lines containing O into themselves, circles not containing Ointo circles not containing O, circles containing O into lines not containingO.2.3.7 Geometric InequalitiesTheorem 2.89 (The triangle inequality). For any three points A, B, Cin a plane AB + BC ≥ AC. Equality occurs when A, B, C are collinear andB(A, B, C).
2.5 Combinatorics 23Definition 2.141. A permutation of order n is a bijection of {1, 2, . . . , n}into itself (a special case of variation for k = n). For a given n the number ofdifferent permutations is Pn = n!.Definition 2.142. A combination of order n over k is a k-element subset of{1, 2, . . . , n}. For a given n and k the number of different combinations is Cn = n . k kDefinition 2.143. A permutation with repetition of order n is a bijection of{1, 2, . . . , n} into a multiset of n elements. A multiset is defined to be a set inwhich certain elements are deemed mutually indistinguishable (for example,as in {1, 1, 2, 3}). If {1, 2 . . . , s} denotes a set of different elements in the multiset and theelement i appears αi times in the multiset, then number of different permuta-tions with repetition is Pn,α1 ,...,αs = α1 !·αn! s ! . A combination is a special 2 !···αcase of permutation with repetition for a multiset with two different elements.Theorem 2.144 (The pigeonhole principle). If a set of nk + 1 differ-ent elements is partitioned into n mutually disjoint subsets, then at least onesubset will contain at least k + 1 elements.Theorem 2.145 (The inclusion–exclusion principle). Let S1 , S2 , . . . , Snbe a family of subsets of the set S. The number of elements of S contained innone of the subsets is given by the formula n |S(S1 ∪ · · · ∪ Sn )| = |S| − (−1)k |Si1 ∩ · · · ∩ Sik | . k=1 1≤i1 <···<ik ≤n2.5.2 Graph TheoryDefinition 2.146. A graph G = (V, E) is a set of objects, i.e., vertices, Vpaired with the multiset E of some pairs of elements of V , i.e., edges. When(x, y) ∈ E, for x, y ∈ V , the vertices x and y are said to be connected by anedge; i.e., the vertices are the endpoints of the edge. A graph for which the multiset E reduces to a proper set (i.e., the verticesare connected by at most one edge) and for which no vertex is connected toitself is called a proper graph. A finite graph is one in which |E| and |V | are finite.Definition 2.147. An oriented graph is one in which the pairs in E are or-dered.Definition 2.148. A proper graph Kn containing n vertices and in whicheach pair of vertices is connected is called a complete graph.
24 2 Basic Concepts and FactsDefinition 2.149. A k-partite graph (bipartite for k = 2) Ki1 ,i2 ,...,ik is a graphwhose set of vertices V can be partitioned into k non-empty disjoint subsetsof cardinalities i1 , i2 , . . . , ik such that each vertex x in a subset W of V isconnected only with the vertices not in W .Definition 2.150. The degree d(x) of a vertex x is the number of times x isthe endpoint of an edge (thus, self-connecting edges are counted twice). Anisolated vertex is one with the degree 0.Theorem 2.151. For a graph G = (V, E) the following identity holds: d(x) = 2|E|. x∈VAs a consequence, the number of vertices of odd degree is even.Definition 2.152. A trajectory (path) of a graph is a finite sequence of ver-tices, each connected to the previous one. The length of a trajectory is thenumber of edges through which it passes. A circuit is a path that ends in thestarting vertex. A cycle is a circuit in which no vertex appears more than once(except the initial/final vertex). A graph is connected if there exists a trajectory between any two vertices.Definition 2.153. A subgraph G = (V , E ) of a graph G = (V, E) is agraph such that V ⊆ V and E contains exactly the edges of E connectingpoints in V . A connected component of a graph is a connected subgraph suchthat no vertex of the component is connected with any vertex outside of thecomponent.Definition 2.154. A tree is a connected graph that contains no cycles.Theorem 2.155. A tree with n vertices has exactly n − 1 edges and at leasttwo vertices of degree 1.Definition 2.156. An Euler path is a path in which each edge appears exactlyonce. Likewise, an Euler circuit is an Euler path that is also a circuit.Theorem 2.157. The following conditions are necessary and sufficient for afinite connected graph G to have an Euler path: • If each vertex has even degree, then the graph contains an Euler circuit. • If all vertices except two have even degree, then the graph contains an Euler path that is not a circuit (it starts and ends in the two odd vertices).Definition 2.158. A Hamilton circuit is a circuit that contains each vertexof G exactly once (trivially, it is also a cycle). A simple rule to determine whether a graph contains a Hamilton circuithas not yet been discovered.
3Problems3.1 The First IMOBucharest–Brasov, Romania, July 23–31, 19593.1.1 Contest Problems First Day 21n+41. (POL) For every integer n prove that the fraction 14n+3 cannot be reduced any further.2. (ROM) For which real numbers x do the following equations hold: √ √ √ (a) x + 2x − 1 + x + 2x − 1 = 2 , √ √ (b) x + 2x − 1 + x + 2x − 1 = 1 , √ √ (c) x + 2x − 1 + x + 2x − 1 = 2 ?3. (HUN) Let x be an angle and let the real numbers a, b, c, cos x satisfy the following equation: a cos2 x + b cos x + c = 0 . Write the analogous quadratic equation for a, b, c, cos 2x. Compare the given and the obtained equality for a = 4, b = 2, c = −1. Second Day4. (HUN) Construct a right-angled triangle whose hypotenuse c is given if it is known that the median from the right angle equals the geometric mean of the remaining two sides of the triangle.5. (ROM) A segment AB is given and on it a point M . On the same side of AB squares AM CD and BM F E are constructed. The circumcircles of the two squares, whose centers are P and Q, intersect in M and another point N .
28 3 Problems (a) Prove that lines F A and BC intersect at N . (b) Prove that all such constructed lines M N pass through the same point S, regardless of the selection of M . (c) Find the locus of the midpoints of all segments P Q, as M varies along the segment AB.6. (CZS) Let α and β be two planes intersecting at a line p. In α a point A is given and in β a point C is given, neither of which lies on p. Construct B in α and D in β such that ABCD is an equilateral trapezoid, AB CD, in which a circle can be inscribed.
3.2 IMO 1960 293.2 The Second IMOBucharest–Sinaia, Romania, July 18–25, 19603.2.1 Contest Problems First Day1. (BUL) Find all the three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the digits of the initial number.2. (HUN) For which real numbers x does the following inequality hold: 4x2 √ < 2x + 9 ? (1 − 1 + 2x)23. (ROM) A right-angled triangle ABC is given for which the hypotenuse BC has length a and is divided into n equal segments, where n is odd. Let α be the angle with which the point A sees the segment containing the middle of the hypotenuse. Prove that 4nh tan α = , (n2 − 1)a where h is the height of the triangle. Second Day4. (HUN) Construct a triangle ABC whose lengths of heights ha and hb (from A and B, respectively) and length of median ma (from A) are given.5. (CZS) A cube ABCDA B C D is given. (a) Find the locus of all midpoints of segments XY , where X is any point on segment AC and Y any point on segment B D . −−→ −→ − (b) Find the locus of all points Z on segments XY such that ZY = 2XZ.6. (BUL) An isosceles trapezoid with bases a and b and height h is given. (a) On the line of symmetry construct the point P such that both (non- base) sides are seen from P with an angle of 90◦ . (b) Find the distance of P from one of the bases of the trapezoid. (c) Under what conditions for a, b, and h can the point P be constructed (analyze all possible cases)?7. (GDR) A sphere is inscribed in a regular cone. Around the sphere a cylinder is circumscribed so that its base is in the same plane as the base of the cone. Let V1 be the volume of the cone and V2 the volume of the cylinder. (a) Prove that V1 = V2 is impossible. (b) Find the smallest k for which V1 = kV2 , and in this case construct the angle at the vertex of the cone.
30 3 Problems3.3 The Third IMOBudapest–Veszprem, Hungary, July 6–16, 19613.3.1 Contest Problems First Day1. (HUN) Solve the following system of equations: x + y + z = a, x + y 2 + z 2 = b2 , 2 xy = z 2 , where a and b are given real numbers. What conditions must hold on a and b for the solutions to be positive and distinct?2. (POL) Let a, b, and c be the lengths of a triangle whose area is S. Prove that √ a2 + b2 + c2 ≥ 4S 3 . In what case does equality hold?3. (BUL) Solve the equation cosn x− sinn x = 1, where n is a given positive integer. Second Day4. (GDR) In the interior of P1 P2 P3 a point P is given. Let Q1 , Q2 , and Q3 respectively be the intersections of P P1 , P P2 , and P P3 with the opposing edges of P1 P2 P3 . Prove that among the ratios P P1 /P Q1 , P P2 /P Q2 , and P P3 /P Q3 there exists at least one not larger than 2 and at least one not smaller than 2.5. (CZS) Construct a triangle ABC if the following elements are given: AC = b, AB = c, and AM B = ω (ω < 90o ), where M is the midpoint of BC. Prove that the construction has a solution if and only if ω b tan ≤c<b. 2 In what case does equality hold?6. (ROM) A plane is given and on one side of the plane three noncollinear points A, B, and C such that the plane determined by them is not parallel to . Three arbitrary points A , B , and C in are selected. Let L, M , and N be the midpoints of AA , BB , and CC , and G the centroid of LM N . Find the locus of all points obtained for G as A , B , and C are varied (independently of each other) across .
3.4 IMO 1962 313.4 The Fourth IMOPrague–Hluboka, Czechoslovakia, July 7–15, 19623.4.1 Contest Problems First Day1. (POL) Find the smallest natural number n with the following properties: (a) In decimal representation it ends with 6. (b) If we move this digit to the front of the number, we get a number 4 times larger.2. (HUN) Find all real numbers x for which √ √ 1 3−x− x+1> . 23. (CZS) A cube ABCDA B C D is given. The point X is moving at a constant speed along the square ABCD in the direction from A to B. The point Y is moving with the same constant speed along the square BCC B in the direction from B to C . Initially, X and Y start out from A and B respectively. Find the locus of all the midpoints of XY . Second Day4. (ROM) Solve the equation cos2 x + cos2 2x + cos2 3x = 1 .5. (BUL) On the circle k three points A, B, and C are given. Construct the fourth point on the circle D such that one can inscribe a circle in ABCD.6. (GDR) Let ABC be an isosceles triangle with circumradius r and inra- dius ρ. Prove that the distance d between the circumcenter and incenter is given by d = r(r − 2ρ) .7. (USS) Prove that a tetrahedron SABC has five different spheres that touch all six lines determined by its edges if and only if it is regular.
32 3 Problems3.5 The Fifth IMOWroclaw, Poland, July 5–13, 19633.5.1 Contest Problems First Day √1. (CZS) Determine all real solutions of the equation x2 − p+2 x2 − 1 = x, where p is a real number.2. (USS) Find the locus of points in space that are vertices of right angles of which one ray passes through a given point and the other intersects a given segment.3. (HUN) Prove that if all the angles of a convex n-gon are equal and the lengths of consecutive edges a1 , . . . , an satisfy a1 ≥ a2 ≥ · · · ≥ an , then a1 = a2 = · · · = a n . Second Day4. (USS) Find all solutions x1 , . . . , x5 to the system of equations ⎧ ⎪ x5 + x2 = yx1 , ⎪ ⎪ ⎪ x1 + x3 = yx2 , ⎨ x2 + x4 = yx3 , ⎪ ⎪ x3 + x5 = yx4 , ⎪ ⎪ ⎩ x4 + x1 = yx5 , where y is a real parameter.5. (GDR) Prove that cos π − cos 2π + cos 3π = 1 . 7 7 7 26. (HUN) Five students A, B, C, D, and E have taken part in a certain competition. Before the competition, two persons X and Y tried to guess the rankings. X thought that the ranking would be A, B, C, D, E; and Y thought that the ranking would be D, A, E, C, B. At the end, it was revealed that X didn't guess correctly any rankings of the participants, and moreover, didn't guess any of the orderings of pairs of consecutive participants. On the other hand, Y guessed the correct rankings of two participants and the correct ordering of two pairs of consecutive partici- pants. Determine the rankings of the competition.
3.6 IMO 1964 333.6 The Sixth IMOMoscow, Soviet Union, June 30–July 10, 19643.6.1 Contest Problems First Day1. (CZS) (a) Find all natural numbers n such that the number 2n − 1 is divisible by 7. (b) Prove that for all natural numbers n the number 2n + 1 is not divisible by 7.2. (HUN) Denote by a, b, c the lengths of the sides of a triangle. Prove that a2 (b + c − a) + b2 (c + a − b) + c2 (a + b − c) ≤ 3abc.3. (YUG) The incircle is inscribed in a triangle ABC with sides a, b, c. Three tangents to the incircle are drawn, each of which is parallel to one side of the triangle ABC. These tangents form three smaller triangles (internal to ABC) with the sides of ABC. In each of these triangles an incircle is inscribed. Determine the sum of areas of all four incircles. Second Day4. (HUN) Each of 17 students talked with every other student. They all talked about three different topics. Each pair of students talked about one topic. Prove that there are three students that talked about the same topic among themselves.5. (ROM) Five points are given in the plane. Among the lines that connect these five points, no two coincide and no two are parallel or perpendicular. Through each point we construct an altitude to each of the other lines. What is the maximal number of intersection points of these altitudes (excluding the initial five points)?6. (POL) Given a tetrahedron ABCD, let D1 be the centroid of the triangle ABC and let A1 , B1 , C1 be the intersection points of the lines parallel to DD1 and passing through the points A, B, C with the opposite faces of the tetrahedron. Prove that the volume of the tetrahedron ABCD is one- third the volume of the tetrahedron A1 B1 C1 D1 . Does the result remain true if the point D1 is replaced with any point inside the triangle ABC?
34 3 Problems3.7 The Seventh IMOBerlin, DR Germany, July 3–13, 19653.7.1 Contest Problems First Day1. (YUG) Find all real numbers x ∈ [0, 2π] such that √ √ √ 2 cos x ≤ | 1 + sin 2x − 1 − sin 2x| ≤ 2.2. (POL) Consider the system of equations ⎧ ⎨ a11 x1 + a12 x2 + a13 x3 = 0, a21 x1 + a22 x2 + a23 x3 = 0, ⎩ a31 x1 + a32 x2 + a33 x3 = 0, whose coefficients satisfy the following conditions: (a) a11 , a22 , a33 are positive real numbers; (b) all other coefficients are negative; (c) in each of the equations the sum of the coefficients is positive. Prove that x1 = x2 = x3 = 0 is the only solution to the system.3. (CZS) A tetrahedron ABCD is given. The lengths of the edges AB and CD are a and b, respectively, the distance between the lines AB and CD is d, and the angle between them is equal to ω. The tetrahedron is divided into two parts by the plane π parallel to the lines AB and CD. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane π from AB and CD is equal to k. Second Day4. (USS) Find four real numbers x1 , x2 , x3 , x4 such that the sum of any of the numbers and the product of other three is equal to 2.5. (ROM) Given a triangle OAB such that ∠AOB = α < 90◦ , let M be an arbitrary point of the triangle different from O. Denote by P and Q the feet of the perpendiculars from M to OA and OB, respectively. Let H be the orthocenter of the triangle OP Q. Find the locus of points H when: (a) M belongs to the segment AB; (b) M belongs to the interior of OAB.6. (POL) We are given n ≥ 3 points in the plane. Let d be the maximal distance between two of the given points. Prove that the number of pairs of points whose distance is equal to d is less than or equal to n.
3.8 IMO 1966 353.8 The Eighth IMOSofia, Bulgaria, July 3–13, 19663.8.1 Contest Problems First Day1. (USS) Three problems A, B, and C were given on a mathematics olympiad. All 25 students solved at least one of these problems. The num- ber of students who solved B and not A is twice the number of students who solved C and not A. The number of students who solved only A is greater by 1 than the number of students who along with A solved at least one other problem. Among the students who solved only one problem, half solved A. How many students solved only B?2. (HUN) If a, b, and c are the sides and α, β, and γ the respective angles of the triangle for which a + b = tan γ (a tan α + b tan β), prove that the 2 triangle is isosceles.3. (BUL) Prove that the sum of distances from the center of the circum- sphere of the regular tetrahedron to its four vertices is less than the sum of distances from any other point to the four vertices. Second Day4. (YUG) Prove the following equality: 1 1 1 1 + + + ···+ = cot x − cot 2n x, sin 2x sin 4x sin 8x sin 2n x where n ∈ N and x ∈ πZ/2k for every k ∈ N. /5. (CZS) Solve the following system of equations: |a1 − a2 |x2 + |a1 − a3 |x3 + |a1 − a4 |x4 = 1, |a2 − a1 |x1 + |a2 − a3 |x3 + |a2 − a4 |x4 = 1, |a3 − a1 |x1 + |a3 − a2 |x2 + |a3 − a4 |x4 = 1, |a4 − a1 |x1 + |a4 − a2 |x2 + |a4 − a3 |x3 = 1, where a1 , a2 , a3 , and a4 are mutually distinct real numbers.6. (POL) Let M , K, and L be points on (AB), (BC), and (CA), respec- tively. Prove that the area of at least one of the three triangles M AL, KBM , and LCK is less than or equal to one-fourth the area of ABC.
36 3 Problems3.8.2 Some Longlisted Problems 1959–19661. (CZS) We are given n > 3 points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?2. (GDR) Given n positive real numbers a1 , a2 , . . . , an such that a1 a2 · · · an = 1, prove that (1 + a1 )(1 + a2 ) · · · (1 + an ) ≥ 2n .3. (BUL) A regular triangular prism has height h and a base of side length a. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the light enters and the length of its path inside the prism.4. (POL) Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.5. (USS) Prove the inequality π sin x π cos x tan + tan >1 4 sin α 4 cos α for any x, α with 0 ≤ x ≤ π/2 and π/6 < y < π/3.6. (USS) A convex planar polygon M with perimeter l and area S is given. Let M (R) be the set of all points in space that lie a distance at most R from a point of M. Show that the volume V (R) of this set equals 4 3 π 2 V (R) = πR + lR + 2SR. 3 27. (USS) For which arrangements of two infinite circular cylinders does their intersection lie in a plane?8. (USS) We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?9. (ROM) Find x such that sin 3x cos(60◦ − 4x) + 1 = 0, sin(60◦ − 7x) − cos(30◦ + x) + m where m is a fixed real number.10. (GDR) How many real solutions are there to the equation x = 1964 sin x − 189?
3.8 IMO 1966 3711. (CZS) Does there exist an integer z that can be written in two different ways as z = x! + y!, where x, y are natural numbers with x ≤ y?12. (BUL) Find digits x, y, z such that the equality √ xx · · · x − yy · · · y = zz · · · z 2n n n holds for at least two values of n ∈ N, and in that case find all n for which this equality is true.13. (YUG) Let a1 , a2 , . . . , an be positive real numbers. Prove the inequality ⎛ ⎞2 n 1 1 ⎠ ≥ 4⎝ 2 i<j ai aj i<j ai + aj and find the conditions on the numbers ai for equality to hold.14. (POL) Compute the largest number of regions into which one can divide a disk by joining n points on its circumference.15. (POL) Points A, B, C, D lie on a circle such that AB is a diameter and CD is not. If the tangents at C and D meet at P while AC and BD meet at Q, show that P Q is perpendicular to AB.16. (CZS) We are given a circle K with center S and radius 1 and a square Q with center M and side 2. Let XY be the hypotenuse of an isosceles right triangle XY Z. Describe the locus of points Z as X varies along K and Y varies along the boundary of Q.17. (ROM) Suppose ABCD and A B C D are two parallelograms arbi- trarily arranged in space, and let points M, N, P, Q divide the segments AA , BB , CC , DD respectively in equal ratios. (a) Show that M N P Q is a parallelogram; (b) Find the locus of M N P Q as M varies along the segment AA .18. (HUN) Solve the equation sin x + cos x = 1 , where p is a real parameter. 1 1 p Discuss for which values of p the equation has at least one real solution and determine the number of solutions in [0, 2π) for a given p.19. (HUN) Construct a triangle given the three exradii.20. (HUN) We are given three equal rectangles with the same center in three mutually perpendicular planes, with the long sides also mutually perpendicular. Consider the polyhedron with vertices at the vertices of these rectangles. (a) Find the volume of this polyhedron; (b) can this polyhedron be regular, and under what conditions?21. (BUL) Prove that the volume V and the lateral area S of a right circular 2 3 cone satisfy the inequality 6V π ≤ 2S √ π 3 . When does equality occur?
38 3 Problems22. (BUL) Assume that two parallelograms P, P of equal areas have sides a, b and a , b respectively such that a ≤ a ≤ b ≤ b and a segment of length b can be placed inside P . Prove that P and P can be partitioned into four pairwise congruent parts.23. (BUL) Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. (a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. (b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.24. (POL) There are n ≥ 2 people in a room. Prove that there exist two among them having equal numbers of friends in that room. (Friendship is always mutual.) √ √ √25. (GDR) Show that tan 7◦ 30 = 6 + 2 − 3 − 2.26. (CZS) (a) Prove that (a1 + a2 + · · · + ak )2 ≤ k(a2 + · · · + a2 ), where k ≥ 1 1 k is a natural number and a1 , . . . , ak are arbitrary real numbers. (b) If real numbers a1 , . . . , an satisfy a1 + a2 + · · · + a n ≥ (n − 1)(a2 + · · · + a2 ), 1 n show that they are all nonnegative.27. (GDR) We are given a circle K and a point P lying on a line g. Construct a circle that passes through P and touches K and g.28. (CZS) Let there be given a circle with center S and radius 1 in the plane, and let ABC be an arbitrary triangle circumscribed about the circle such that SA ≤ SB ≤ SC. Find the loci of the vertices A, B, C.29. (ROM) (a) Find the number of ways 500 can be represented as a sum of consecutive integers. (b) Find the number of such representations for N = 2α 3β 5γ , α, β, γ ∈ N. Which of these representations consist only of natural numbers? (c) Determine the number of such representations for an arbitrary natural number N .30. (ROM) If n is a natural number, prove that 3 (a) log10 (n + 1) > 10n + log10 n; (b) log n! > 10 2 + 1 + · · · + n − 1 . 3n 1 3 131. (ROM) Solve the equation |x2 − 1| + |x2 − 4| = mx as a function of the parameter m. Which pairs (x, m) of integers satisfy this equation?32. (BUL) The sides a, b, c of a triangle ABC form an arithmetic progression; the sides of another triangle A1 B1 C1 also form an arithmetic progression.
3.8 IMO 1966 39 Suppose that ∠A = ∠A1 . Prove that the triangles ABC and A1 B1 C1 are similar.33. (BUL) Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle one draws segments tangent to the smaller circle. Prove that the length of one of these segments equals the sum of the lengths of the other two.34. (BUL) Determine all pairs of positive integers (x, y) satisfying the equa- tion 2x = 3y + 5.35. (POL) If a, b, c, d are integers such that ad is odd and bc is even, prove that at least one root of the polynomial ax3 + bx2 + cx + d is irrational.36. (POL) Let ABCD be a cyclic quadrilateral. Show that the centroids of the triangles ABC, CDA, BCD, DAB lie on a circle.37. (POL) Prove that the perpendiculars drawn from the midpoints of the sides of a cyclic quadrilateral to the opposite sides meet at one point.38. (ROM) Two concentric circles have radii R and r respectively. Determine the greatest possible number of circles that are tangent to both these circles and mutually nonintersecting. Prove that this number lies between √ √ 3 √R+ r 2 · R−√r − 1 and 63 · R−r . 20 R+r39. (ROM) In a plane, a circle with center O and radius R and two points A, B are given. (a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P , say P1 , P2 , and find the distance between them if OA = a, OB = b, AB = d.40. (CZS) For a positive real number p, find all real solutions to the equation x2 + 2px − p2 − x2 − 2px − p2 = 1.41. (CZS) If A1 A2 . . . An is a regular n-gon (n ≥ 3), how many different obtuse triangles Ai Aj Ak exist?42. (CZS) Let a1 , a2 , . . . , an (n ≥ 2) be a sequence of integers. Show that there is a subsequence ak1 , ak2 , . . . , akm , where 1 ≤ k1 < k2 < · · · < km ≤ n, such that a21 + a22 + · · · + a2m is divisible by n. k k k43. (CZS) Five points in a plane are given, no three of which are collinear. Every two of them are joined by a segment, colored either red or gray, so that no three segments form a triangle colored in one color. (a) Prove that (1) every point is a vertex of exactly two red and two gray segments, and (2) the red segments form a closed path that passes through each point. (b) Give an example of such a coloring.
40 3 Problems44. (YUG) What is the greatest number of balls of radius 1/2 that can be placed within a rectangular box of size 10 × 10 × 1?45. (YUG) An alphabet consists of n letters. What is the maximal length of a word, if (i) two neighboring letters in a word are always different, and (ii) no word abab (a = b) can be obtained by omitting letters from the given word?46. (YUG) Let |b − a| b + a 2 |b − a| b + a 2 f (a, b, c) = + − + + + . |ab| ab c |ab| ab c Prove that f (a, b, c) = 4 max{1/a, 1/b, 1/c}.47. (ROM) Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the triangle. Compute this minimum length in terms of the sides a, b, c of the triangle.48. (USS) Find all positive numbers p for which the equation x2 +px+3p = 0 has integral roots.49. (USS) Two mirror walls are placed to form an angle of measure α. There is a candle inside the angle. How many reflections of the candle can an observer see?50. (USS) Given a quadrangle of sides a, b, c, d and area S, show that S ≤ 2 · 2 . a+c b+d51. (USS) In a school, n children numbered 1 to n are initially arranged in the order 1, 2, . . . , n. At a command, every child can either exchange its position with any other child or not move. Can they rearrange into the order n, 1, 2, . . . , n − 1 after two commands?52. (USS) A figure of area 1 is cut out from a sheet of paper and divided into 10 parts, each of which is colored in one of 10 colors. Then the figure is turned to the other side and again divided into 10 parts (not necessarily in the same way). Show that it is possible to color these parts in the 10 colors so that the total area of the portions of the figure both of whose sides are of the same color is at least 0.1.53. (USS, 1966) Prove that in every convex hexagon of area S one can draw 1 a diagonal that cuts off a triangle of area not exceeding 6 S.54. (USS, 1966) Find the last two digits of a sum of eighth powers of 100 consecutive integers.55. (USS, 1966) Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such that all the angles of the triangle lie in the interval [40◦ , 70◦ ].
3.8 IMO 1966 4156. (USS, 1966) Let ABCD be a tetrahedron such that AB ⊥ CD, AC ⊥ BD, and AD ⊥ BC. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.57. (USS, 1966) Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.
42 3 Problems3.9 The Ninth IMOCetinje, Yugoslavia, July 2–13, 19673.9.1 Contest Problems First Day (July 5)1. ABCD is a parallelogram; AB = a, AD = 1, α is the size of ∠DAB, and the three angles of the triangle ABD are acute. Prove that the four circles KA , KB , KC , KD , each of radius 1, whose centers are the vertices A, B, √ C, D, cover the parallelogram if and only if a ≤ cos α + 3 sin α.2. Exactly one side of a tetrahedron is of length greater than 1. Show that its volume is less than or equal to 1/8.3. Let k, m, and n be positive integers such that m + k + 1 is a prime number greater than n + 1. Write cs for s(s + 1). Prove that the product (cm+1 −ck )(cm+2 −ck ) · · · (cm+n −ck ) is divisible by the product c1 c2 · · · cn . Second Day (July 6)4. The triangles A0 B0 C0 and A B C have all their angles acute. Describe how to construct one of the triangles ABC, similar to A B C and cir- cumscribing A0 B0 C0 (so that A, B, C correspond to A , B , C , and AB passes through C0 , BC through A0 , and CA through B0 ). Among these triangles ABC describe, and prove, how to construct the triangle with the maximum area.5.6. In a sports competition lasting n days there are m medals to be won. On the first day, one medal and 1/7 of the remaining m − 1 medals are won. On the second day, 2 medals and 1/7 of the remainder are won. And so on. On the nth day exactly n medals are won. How many days did the competition last and what was the total number of medals?3.9.2 Longlisted Problems1. (BUL 1) Prove that all numbers in the sequence
44 3 Problems least one decomposition for this n. Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily given positive number.11. (CZS 5) Let n be a positive integer. Find the maximal number of non- congruent triangles whose side lengths are integers less than or equal to n.12. (CZS 6) Given a segment AB of the length 1, define the set M of points in the following way: it contains the two points A, B, and also all points obtained from A, B by iterating the following rule: (∗) for every pair of points X, Y in M , the set M also contains the point Z of the segment XY for which Y Z = 3XZ. (a) Prove that the set M consists of points X from the segment AB for which the distance from the point A is either 3k 3k − 2 AX = or AX = , 4n 4n where n, k are nonnegative integers. (b) Prove that the point X0 for which AX0 = 1/2 = X0 B does not belong to the set M .13. (GDR 1) Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius r there exists one (or more) with maximal area. If so, determine their shape and area.14. (GDR 2) Which fraction p/q, where p, q are positive integers less than √ 100, is closest to 2? Find all digits after the decimal point in the decimal representation of √this fraction that coincide with digits in the decimal representation of 2 (without using any tables).15. (GDR 3) Suppose tan α = p/q, where p and q are integers and q = 0. Prove that the number tan β for which tan 2β = tan 3α is rational only when p2 + q 2 is the square of an integer.16. (GDR 4) Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by numbers of the form zk1 ,k2 = k1 r1 +k2 r2 , k1 , k2 integers; i.e., for every real number x and every positive real number p two integers k1 and k2 can be found such that |x − (k1 r1 + k2 r2 )| < p.17. (GBR 1)IMO3 Let k, m, and n be positive integers such that m + k + 1 is a prime number greater than n + 1. Write cs for s(s + 1). Prove that the product (cm+1 − ck )(cm+2 − ck ) · · · (cm+n − ck ) is divisible by the product c1 c2 · · · c n .18. (GBR 5) If x is a positive rational number, show that x can be uniquely expressed in the form a2 a3 x = a1 + + + ··· , 2! 3!
3.9 IMO 1967 45 where a1 , a2 , . . . are integers, 0 ≤ an ≤ n − 1 for n > 1, and the series terminates. Show also that x can be expressed as the sum of reciprocals of different integers, each of which is greater than 106 .19. (GBR 6) The n points P1 , P2 , . . . , Pn are placed inside or on the bound- ary of a disk of radius 1 in such a way that the minimum distance dn between any two of these points has its largest possible value Dn . Calcu- late Dn for n = 2 to 7 and justify your answer.20. (HUN 1) In space, n points (n ≥ 3) are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain a polygonal line in such a way. 121. (HUN 2) Without using any tables, find the exact value of the product π 2π 3π 4π 5π 6π 7π P = cos cos cos cos cos cos cos . 15 15 15 15 15 15 1522. (HUN 3) The distance between the centers of the circles k1 and k2 with radii r is equal to r. Points A and B are on the circle k1 , symmetric with respect to the line connecting the centers of the circles. Point P is an arbitrary point on k2 . Prove that P A2 + P B 2 ≥ 2r2 . When does equality hold?23. (HUN 4) Prove that for an arbitrary pair of vectors f and g in the plane, the inequality af 2 + bf g + cg 2 ≥ 0 holds if and only if the following conditions are fulfilled: a ≥ 0, c ≥ 0, 4ac ≥ b2 .24. (HUN 5)IMO6 Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer number of coins, find the number of children and the number of coins.25. (HUN 6) Three disks of diameter d are touching a sphere at their centers. Moreover, each disk touches the other two disks. How do we choose the radius R of the sphere so that the axis of the whole figure makes an angle 1 The statement so formulated is false. It would be trivially true under the addi- tional assumption that the polygonal line is closed. However, from the offered solution, which is not clear, it does not seem that the proposer had this in mind.
46 3 Problems of 60◦ with the line connecting the center of the sphere with the point on the disks that is at the largest distance from the axis? (The axis of the figure is the line having the property that rotation of the figure through 120◦ about that line brings the figure to its initial position. The disks are all on one side of the plane, pass through the center of the sphere, and are orthogonal to the axes.)26. (ITA 1) Let ABCD be a regular tetrahedron. To an arbitrary point M on one edge, say CD, corresponds the point P = P (M ), which is the intersection of two lines AH and BK, drawn from A orthogonally to BM and from B orthogonally to AM . What is the locus of P as M varies?27. (ITA 2) Which regular polygons can be obtained (and how) by cutting a cube with a plane?28. (ITA 3) Find values of the parameter u for which the expression tan(x − u) + tan x + tan(x + u) y= tan(x − u) tan x tan(x + u) does not depend on x.29. (ITA 4)IMO4 The triangles A0 B0 C0 and A B C have all their angles acute. Describe how to construct one of the triangles ABC, similar to A B C and circumscribing A0 B0 C0 (so that A, B, C correspond to A , B , C , and AB passes through C0 , BC through A0 , and CA through B0 ). Among these triangles ABC, describe, and prove, how to construct the triangle with the maximum area.30. (MON 1) Given m + n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . . , n), determine the number of pairs (ai , bj ) for which |i − j| ≥ k, where k is a nonnegative integer.31. (MON 2) An urn contains balls of k different colors; there are ni balls of the ith color. Balls are drawn at random from the urn, one by one, without replacement. Find the smallest number of draws necessary for getting m balls of the same color.32. (MON 3) Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are α, β, γ.33. (MON 4) In what case does the system x + y + mz = a, x + my + z = b, mx + y + z = c, have a solution? Find the conditions under which the unique solution of the above system is an arithmetic progression.
3.9 IMO 1967 4734. (MON 5) The faces of a convex polyhedron are six squares and eight equilateral triangles, and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge are equal. Prove that it is possible to circumscribe a sphere around this polyhedron and compute the ratio of the squares of the volumes of the polyhedron and of the ball whose boundary is the circumscribed sphere.35. (MON 6) Prove the identity n n x 2k 1 x tan 1 + 2k k = sec2n + secn x. k 2 1 − tan (x/2) 2 2 k=036. (POL 1) Prove that the center of the sphere circumscribed around a tetrahedron ABCD coincides with the center of a sphere inscribed in that tetrahedron if and only if AB = CD, AC = BD, and AD = BC.37. (POL 2) Prove that for arbitrary positive numbers the following in- equality holds: 1 1 1 a 8 + b 8 + c8 + + ≤ . a b c a 3 b 3 c338. (POL 3) Does there exist an integer such that its cube is equal to 3n2 + 3n + 7, where n is integer?39. (POL 4) Show that the triangle whose angles satisfy the equality sin2 A + sin2 B + sin2 C =2 cos2 A + cos2 B + cos2 C is a right-angled triangle.40. (POL 5)IMO2 Exactly one side of a tetrahedron is of length greater than 1. Show that its volume is less than or equal to 1/8.41. (POL 6) A line l is drawn through the intersection point H of the altitudes of an acute-angled triangle. Prove that the symmetric images la , lb , lc of l with respect to sides BC, CA, AB have one point in common, which lies on the circumcircle of ABC.42. (ROM 1) Decompose into real factors the expression 1 − sin5 x − cos5 x.43. (ROM 2) The equation x5 + 5λx4 − x3 + (λα − 4)x2 − (8λ + 3)x + λα − 2 = 0 is given. (a) Determine α such that the given equation has exactly one root inde- pendent of λ. (b) Determine α such that the given equation has exactly two roots inde- pendent of λ.
3.9 IMO 1967 4951. (SWE 4) A subset S of the set of integers 0, . . . , 99 is said to have property A if it is impossible to fill a crossword puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets S with property A.52. (SWE 5) In the plane a point O and a sequence of points P1 , P2 , P3 , . . . are given. The distances OP1 , OP2 , OP3 , . . . are r1 , r2 , r3 , . . . , where r1 ≤ r2 ≤ r3 ≤ · · · . Let α satisfy 0 < α < 1. Suppose that for every n the distance from the point Pn to any other point of the sequence is greater α than or equal to rn . Determine the exponent β, as large as possible, such that for some C independent of n,2 rn ≥ Cnβ , n = 1, 2, . . . .53. (SWE 6) In making Euclidean constructions in geometry it is permit- ted to use a straightedge and compass. In the constructions considered in this question, no compasses are permitted, but the straightedge is as- sumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the ruler. Then the distance between the parallel lines is equal to the breadth of the straightedge. Carry through the following constructions with such a straightedge. Construct: (a) The bisector of a given angle. (b) The midpoint of a given rectilinear segment. (c) The center of a circle through three given noncollinear points. (d) A line through a given point parallel to a given line.54. (USS 1) Is it possible to put 100 (or 200) points on a wooden cube such that by all rotations of the cube the points map into themselves? Justify your answer.55. (USS 2) Find all x for which for all n, √ 3 sin x + sin 2x + sin 3x + · · · + sin nx ≤ . 256. (USS 3) In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi.57. (USS 4)IMO5 2 This problem is not elementary. The solution offered by the proposer, which is 1 not quite clear and complete, only shows that if such a β exists, then β ≥ 2(1−α) .
50 3 Problems58. (USS 5) A linear binomial l(z) = Az + B with complex coefficients A and B is given. It is known that the maximal value of |l(z)| on the segment −1 ≤ x ≤ 1 (y = 0) of the real line in the complex plane (z = x + iy) is equal to M . Prove that for every z |l(z)| ≤ M ρ, where ρ is the sum of distances from the point P = z to the points Q1 : z = 1 and Q3 : z = −1.59. (USS 6) On the circle with center O and radius 1 the point A0 is fixed and points A1 , A2 , . . . , A999 , A1000 are distributed in such a way that ∠A0 OAk = k (in radians). Cut the circle at points A0 , A1 , . . . , A1000 . How many arcs with different lengths are obtained?
52 3 Problems3.10.2 Shortlisted Problems1. (SWE 2) Two ships sail on the sea with constant speeds and fixed directions. It is known that at 9:00 the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance?2. (ROM 5)IMO1 Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others.3. (POL 4)IMO4 Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle.4. (BUL 2)IMO3 Let a, b, c be real numbers. Prove that the system of equa- tions ⎧ ⎪ ⎪ ax2 + bx1 + c = x2 , ⎪ ⎪ 1 ⎨ ax2 + bx2 + c = x3 , 2 ············ ⎪ 2 ⎪ axn−1 + bxn−1 + c = xn , ⎪ ⎪ ⎩ ax2 + bxn + c = x1 , n has a unique real solution if and only if (b − 1)2 − 4ac = 0. Remark. It is assumed that a = 0.5. (BUL 5) Let hn be the apothem (distance from the center to one of the sides) of a regular n-gon (n ≥ 3) inscribed in a circle of radius r. Prove the inequality (n + 1)hn+1 − nhn > r. Also prove that if r on the right side is replaced with a greater number, the inequality will not remain true for all n ≥ 3.6. (HUN 1) If ai (i = 1, 2, . . . , n) are distinct non-zero real numbers, prove that the equation a1 a2 an + + ··· + =n a1 − x a2 − x an − x has at least n − 1 real roots.7. (HUN 5) Prove that the product of the radii of three circles exscribed to √ a given triangle does not exceed 3 8 3 times the product of the side lengths of the triangle. When does equality hold?8. (ROM 2) Given an oriented line ∆ and a fixed point A on it, consider all trapezoids ABCD one of whose bases AB lies on ∆, in the positive direction. Let E, F be the midpoints of AB and CD respectively. Find the loci of vertices B, C, D of trapezoids that satisfy the following: (i) |AB| ≤ a (a fixed); (ii) |EF | = l (l fixed);
3.10 IMO 1968 53 (iii) the sum of squares of the nonparallel sides of the trapezoid is constant. Remark. The constants are chosen so that such trapezoids exist. 9. (ROM 3) Let ABC be an arbitrary triangle and M a point inside it. Let da , db , dc be the distances from M to sides BC, CA, AB; a, b, c the lengths of the sides respectively, and S the area of the triangle ABC. Prove the inequality 4S 2 abda db + bcdb dc + cadc da ≤ . 3 Prove that the left-hand side attains its maximum when M is the centroid of the triangle.10. (ROM 4) Consider two segments of length a, b (a > b) and a segment √ of length c = ab. (a) For what values of a/b can these segments be sides of a triangle? (b) For what values of a/b is this triangle right-angled, obtuse-angled, or acute-angled?11. (ROM 6) Find all solutions (x1 , x2 , . . . , xn ) of the equation 1 x1 + 1 (x1 + 1)(x2 + 1) (x1 + 1) · · · (xn−1 + 1) 1+ + + + ···+ = 0. x1 x1 x2 x1 x2 x3 x1 x2 · · · xn12. (POL 1) If a and b are arbitrary positive real numbers and m an integer, prove that m a m b 1+ + 1+ ≥ 2m+1 . b a13. (POL 5) Given two congruent triangles A1 A2 A3 and B1 B2 B3 (Ai Ak = Bi Bk ), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.14. (BUL 5) A line in the plane of a triangle ABC intersects the sides AB and AC respectively at points X and Y such that BX = CY . Find the locus of the center of the circumcircle of triangle XAY .15. (GBR 1)IMO6 Let [x] denote the integer part of x, i.e., the greatest integer not exceeding x. If n is a positive integer, express as a simple function of n the sum n+1 n+2 n + 2i + + ···+ + ··· . 2 4 2i+116. (GBR 3) A polynomial p(x) = a0 xk + a1 xk−1 + · · · + ak with integer coefficients is said to be divisible by an integer m if p(x) is divisible by m for all integers x. Prove that if p(x) is divisible by m, then k!a0 is also divisible by m. Also prove that if a0 , k, m are nonnegative integers for which k!a0 is divisible by m, there exists a polynomial p(x) = a0 xk + · · · + ak divisible by m.
54 3 Problems17. (GBR 4) Given a point O and lengths x, y, z, prove that there exists an equilateral triangle ABC for which OA = x, OB = y, OC = z, if and only if x + y ≥ z, y + z ≥ x, z + x ≥ y (the points O, A, B, C are coplanar).18. (ITA 2) If an acute-angled triangle ABC is given, construct an equilat- eral triangle A B C in space such that lines AA , BB , CC pass through a given point.19. (ITA 5) We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances 0, 1, 2, . . . from it we obtain points with abscisas n = 0, 1, 2, . . . respectively. How many points among them should we take to ensure that some two of them are less than the distance 1/5 apart?20. (CZS 1) Given n (n ≥ 3) points in space such that every three of them form a triangle with one angle greater than or equal to 120◦ , prove that these points can be denoted by A1 , A2 , . . . , An in such a way that for each i, j, k, 1 ≤ i < j < k ≤ n, angle Ai Aj Ak is greater than or equal to 120◦.21. (CZS 2) Let a0 , a1 , . . . , ak (k ≥ 1) be positive integers. Find all positive integers y such that a0 | y; (a0 + a1 ) | (y + a1 ); . . . ; (a0 + an ) | (y + an ).22. (CZS 3)IMO2 Find all positive integers x for which p(x) = x2 − 10x − 22, where p(x) denotes the product of the digits of x.23. (CZS 4) Find all complex numbers m such that polynomial x3 + y 3 + z 3 + mxyz can be represented as the product of three linear trinomials.24. (MON 1) Find the number of all n-digit numbers for which some fixed digit stands only in the ith (1 < i < n) place and the last j digits are distinct.325. (MON 2) Given k parallel lines and a few points on each of them, find the number of all possible triangles with vertices at these given points.426. (GDR)IMO5 Let a > 0 be a real number and f (x) a real function defined on all of R, satisfying for all x ∈ R, 1 f (x + a) = + f (x) − f (x)2 . 2 (a) Prove that the function f is periodic; i.e., there exists b > 0 such that for all x, f (x + b) = f (x). (b) Give an example of such a nonconstant function for a = 1. 3 The problem is unclear. Presumably n, i, j and the ith digit are fixed. 4 The problem is unclear. The correct formulation could be the following: Given k parallel lines l1 , . . . , lk and ni points on the line li , i = 1, 2, . . . , k, find the maximum possible number of triangles with vertices at these points.
3.11 IMO 1969 553.11 The Eleventh IMOBucharest, Romania, July 5–20, 19693.11.1 Contest Problems First Day (July 10)1. Prove that there exist infinitely many natural numbers a with the following property: the number z = n4 + a is not prime for any natural number n.2. Let a1 , a2 , . . . , an be real constants and cos(a2 + x) cos(a3 + x) cos(an + x) y(x) = cos(a1 + x) + + 2 + ···+ . 2 2 2n−1 If x1 , x2 are real and y(x1 ) = y(x2 ) = 0, prove that x1 − x2 = mπ for some integer m.3. Find conditions on the positive real number a such that there exists a tetrahedron k of whose edges (k = 1, 2, 3, 4, 5) have length a, and the other 6 − k edges have length 1. Second Day (July 11)4. Let AB be a diameter of a circle γ. A point C different from A and B5. Given n points in the plane such that no three of them are collinear, prove that one can find at least n−3 convex quadrilaterals with their vertices 2 at these points. 2 26. Under the conditions x1 , x2 > 0, x1 y1 > z1 , and x2 y2 > z2 , prove the inequality 8 1 1 ≤ 2 + x y − z2 . (x1 + x2 )(y1 + y2 ) − (z1 + z2 )2 x1 y1 − z1 2 2 23.11.2 Longlisted Problems1. (BEL 1) A parabola P1 with equation x2 − 2py = 0 and parabola P2 with equation x2 + 2py = 0, p > 0, are given. A line t is tangent to P2 . Find the locus of pole M of the line t with respect to P1 .2. (BEL 2) (a) Find the equations of regular hyperbolas passing through the points A(α, 0), B(β, 0), and C(0, γ). (b) Prove that all such hyperbolas pass through the orthocenter H of the triangle ABC.
56 3 Problems (c) Find the locus of the centers of these hyperbolas. (d) Check whether this locus coincides with the nine-point circle of the triangle ABC. 3. (BEL 3) Construct the circle that is tangent to three given circles. 4. (BEL 4) Let O be a point on a nondegenerate conic. A right angle with vertex O intersects the conic at points A and B. Prove that the line AB passes through a fixed point located on the normal to the conic through the point O. 5. (BEL 5) Let G be the centroid of the triangle OAB. (a) Prove that all conics passing through the points O, A, B, G are hyper- bolas. (b) Find the locus of the centers of these hyperbolas. 6. (BEL 6) Evaluate (cos(π/4) + i sin(π/4))10 in two different ways and prove that 10 10 1 10 − + = 24 . 1 3 2 5 7. (BUL 1) Prove that the equation x3 + y 3 + z 3 = 1969 has no integral solutions. 8. (BUL 2) Find all functions f defined for all x that satisfy the condition xf (y) + yf (x) = (x + y)f (x)f (y), for all x and y. Prove that exactly two of them are continuous. 9. (BUL 3) One hundred convex polygons are placed on a square with edge of length 38 cm. The area of each of the polygons is smaller than π cm2 , and the perimeter of each of the polygons is smaller than 2π cm. Prove that there exists a disk with radius 1 in the square that does not intersect any of the polygons.10. (BUL 4) Let M be the point inside the right-angled triangle ABC (∠C = 90◦ ) such that ∠M AB = ∠M BC = ∠M CA = ϕ. Let ψ be the acute angle between the medians of AC and BC. Prove that sin(ϕ+ψ) sin(ϕ−ψ) = 5.11. (BUL 5) Let Z be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of Z. Let us call such a pair of points unjoinable. Prove that for each real r > 0 there exists an unjoinable pair of points separated by distance r.12. (CZS 1) Given a unit cube, find the locus of the centroids of all tetra- hedra whose vertices lie on the sides of the cube.
3.11 IMO 1969 5713. (CZS 2) Let p be a prime odd number. Is it possible to find p − 1 natural numbers n + 1, n + 2, . . . , n + p − 1 such that the sum of the squares of these numbers is divisible by the sum of these numbers?14. (CZS 3) Let a and b be two positive real numbers. If x is a real solution of the equation x2 + px + q = 0 with real coefficients p and q such that |p| ≤ a, |q| ≤ b, prove that 1 |x| ≤ a+ a2 + 4b . (1) 2 Conversely, if x satisfies (1), prove that there exist real numbers p and q with |p| ≤ a, |q| ≤ b such that x is one of the roots of the equation x2 + px + q = 0.15. (CZS 4) Let K1 , . . . , Kn be nonnegative integers. Prove that K1 !K2 ! · · · Kn ! ≥ [K/n]!n , where K = K1 + · · · + Kn .16. (CZS 5) A convex quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d and angles α = ∠DAB, β = ∠ABC, γ = ∠BCD, and δ = ∠CDA is given. Let s = (a + b + c + d)/2 and P be the area of the quadrilateral. Prove that α+γ P 2 = (s − a)(s − b)(s − c)(s − d) − abcd cos2 . 217. (CZS 6) Let d and p be two real numbers. Find the first term of an arith- metic progression a1 , a2 , a3 , . . . with difference d such that a1 a2 a3 a4 = p. Find the number of solutions in terms of d and p.18. (FRA 1) Let a and b be two nonnegative integers. Denote by H(a, b) the set of numbers n of the form n = pa + qb, where p and q are positive integers. Determine H(a) = H(a, a). Prove that if a = b, it is enough to know all the sets H(a, b) for coprime numbers a, b in order to know all the sets H(a, b). Prove that in the case of coprime numbers a and b, H(a, b) contains all numbers greater than or equal to ω = (a − 1)(b − 1) and also ω/2 numbers smaller than ω.19. (FRA 2) Let n be an integer that is not divisible by any square greater than 1. Denote by xm the last digit of the number xm in the number system with base n. For which integers x is it possible for xm to be 0? Prove that the sequence xm is periodic with period t independent of x. For which x do we have xt = 1. Prove that if m and x are relatively prime, then 0m , 1m , . . . , (n − 1)m are different numbers. Find the minimal period t in terms of n. If n does not meet the given condition, prove that it is possible to have xm = 0 = x1 and that the sequence is periodic starting only from some number k > 1.
58 3 Problems20. (FRA 3) A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is S. If I is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number T = 2S − B − 2I + 2.21. (FRA 4) A right-angled triangle OAB has its right angle at the point B. An arbitrary circle with center on the line OB is tangent to the line OA. Let AT be the tangent to the circle different from OA (T is the point of tangency). Prove that the median from B of the triangle OAB intersects AT at a point M such that M B = M T .22. (FRA 5) Let α(n) be the number of pairs (x, y) of integers such that x + y = n, 0 ≤ y ≤ x, and let β(n) be the number of triples (x, y, z) such that x + y + z = n and 0 ≤ z ≤ y ≤ x. Find a simple relation between α(n) and the integer part of the number n+2 and the relation among β(n), 2 β(n − 3) and α(n). Then evaluate β(n) as a function of the residue of n modulo 6. What can be said about β(n) and 1 + n(n+6) ? And what about 12 (n+3)2 6 ? Find the number of triples (x, y, z) with the property x + y + z ≤ n, 0 ≤ z ≤ y ≤ x as a function of the residue of n modulo 6. What can be said 2 about the relation between this number and the number (n+6)(2n +9n+12) ? 72 −1 b23. (FRA 6) Consider the integer d = a c , where a, b, and c are positive integers and c ≤ a. Prove that the set G of integers that are between 1 and d and relatively prime to d (the number of such integers is denoted by ϕ(d)) can be partitioned into n subsets, each of which consists of b elements. What can be said about the rational number ϕ(d) ? b24. (GBR 1) The polynomial P (x) = a0 xk + a1 xk−1 + · · · + ak , where a0 , . . . , ak are integers, is said to be divisible by an integer m if P (x) is a multiple of m for every integral value of x. Show that if P (x) is divisible by m, then a0 · k! is a multiple of m. Also prove that if a, k, m are positive integers such that ak! is a multiple of m, then a polynomial P (x) with leading term axk can be found that is divisible by m.25. (GBR 2) Let a, b, x, y be positive integers such that a and b have no common divisor greater than 1. Prove that the largest number not ex- pressible in the form ax + by is ab − a − b. If N (k) is the largest number not expressible in the form ax + by in only k ways, find N (k).26. (GBR 3) A smooth solid consists of a right circular cylinder of height h and base-radius r, surmounted by a hemisphere of radius r and center O. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point P on the hemisphere such that OP makes an angle α with
3.11 IMO 1969 59 the horizontal. Show that if α is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through P , show that it will cross the common circular section of the hemisphere and cylinder at a point Q such that ∠SOQ = φ, S being where it initially crossed this section, and sin φ = r tan α . h27. (GBR 4) The segment AB perpendicularly bisects CD at X. Show that, subject to restrictions, there is a right circular cone whose axis passes through X and on whose surface lie the points A, B, C, D. What are the restrictions?28. (GBR 5) Let us define u0 = 0, u1 = 1 and for n ≥ 0, un+2 = aun+1 +bun , a and b being positive integers. Express un as a polynomial in a and b. Prove the result. Given that b is prime, prove that b divides a(ub − 1).29. (GDR 1) Find all real numbers λ such that the equation sin4 x − cos4 x = λ(tan4 x − cot4 x) (a) has no solution, (b) has exactly one solution, (c) has exactly two solutions, (d) has more than two solutions (in the interval (0, π/4)).30. (GDR 2)IMO1 Prove that there exist infinitely many natural numbers a with the following property: The number z = n4 + a is not prime for any natural number n.31. (GDR 3) Find the number of permutations a1 , . . . , an of the set {1, 2, . . . , n} such that |ai − ai+1 | = 1 for all i = 1, 2, . . . , n − 1. Find a recurrence formula and evaluate the number of such permutations for n ≤ 6.32. (GDR 4) Find the maximal number of regions into which a sphere can be partitioned by n circles.33. (GDR 5) Given a ring G in the plane bounded by two concentric circles with radii R and R/2, prove that we can cover this region with 8 disks of radius 2R/5. (A region is covered if each of its points is inside or on the border of some disk.)34. (HUN 1) Let a and b be arbitrary integers. Prove that if k is an integer not divisible by 3, then (a + b)2k + a2k + b2k is divisible by a2 + ab + b2 .35. (HUN 2) Prove that 1 1 1 5 1+ + 3 + ···+ 3 < . 23 3 n 436. (HUN 3) In the plane 4000 points are given such that each line passes through at most 2 of these points. Prove that there exist 1000 disjoint quadrilaterals in the plane with vertices at these points.
60 3 Problems37. (HUN 4)IMO2 If a1 , a2 , . . . , an are real constants, and if y = cos(a1 + x) + 2 cos(a2 + x) + · · · + n cos(an + x) has two zeros x1 and x2 whose difference is not a multiple of π, prove that y ≡ 0.38. (HUN 5) Let r and m (r ≤ m) be natural numbers and Ak = 2k−1 2m π. Evaluate m m 1 sin(rAk ) sin(rAl ) cos(rAk − rAl ). m2 k=1 l=139. (HUN 6) Find the positions of three points A, B, C on the boundary of a unit cube such that min{AB, AC, BC} is the greatest possible.40. (MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.41. (MON 2) Given two numbers x0 and x1 , let α and β be coefficients of the equation 1 − αy − βy 2 = 0. Under the given conditions, find an expression for the solution of the system xn+2 − αxn+1 − βxn = 0, n = 0, 1, 2, . . . .42. (MON 3) Let Ak (1 ≤ k ≤ h) be n-element sets such that each two of them have a nonempty intersection. Let A be the union of all the sets Ak , and let B be a subset of A such that for each k (1 ≤ k ≤ h) the intersection of Ak and B consists of exactly two different elements ak and bk . Find all subsets X of the set A with r elements satisfying the condition that for at least one index k, both elements ak and bk belong to X.43. (MON 4) Let p and q be two prime numbers greater than 3. Prove that if their difference is 2n , then for any two integers m and n, the number S = p2m+1 + q 2m+1 is divisible by 3.44. (MON 5) Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation x2 − ax + b = 0.45. (MON 6)IMO5 Given n points in the plane such that no three of them are collinear, prove that one can find at least n−3 convex quadrilaterals 2 with their vertices at these points.46. (NET 1) The vertices of an (n + 1)-gon are placed on the edges of a regular n-gon so that the perimeter of the n-gon is divided into equal parts. How does one choose these n + 1 points in order to obtain the (n + 1)gon with (a) maximal area;
3.11 IMO 1969 61 (b) minimal area?47. (NET 2)IMO4 Let A and B be points on the circle γ. A point C, different from A and B,48. (NET 3) Let x1 , x2 , x3 , x4 , and x5 be positive integers satisfying x1 +x2 +x3 +x4 +x5 = 1000, x1 −x2 +x3 −x4 +x5 > 0, x1 +x2 −x3 +x4 −x5 > 0, −x1 +x2 +x3 −x4 +x5 > 0, x1 −x2 +x3 +x4 −x5 > 0, −x1 +x2 −x3 +x4 +x5 > 0. (a) Find the maximum of (x1 + x3 )x2 +x4 . (b) In how many different ways can we choose x1 , . . . , x5 to obtain the desired maximum?49. (NET 4) A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by 4.50. (NET 5) The bisectors of the exterior angles of a pentagon B1 B2 B3 B4 B5 form another pentagon A1 A2 A3 A4 A5 . Construct B1 B2 B3 B4 B5 from the given pentagon A1 A2 A3 A4 A5 .51. (NET 6) A curve determined by y= x2 − 10x + 52, 0 ≤ x ≤ 100, is constructed in a rectangular grid. Determine the number of squares cut by the curve.52. (POL 1) Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.53. (POL 2) Given two segments AB and CD not in the same plane, find the locus of points M such that M A2 + M B 2 = M C 2 + M D2 .
62 3 Problems54. (POL 3) Given a polynomial f (x) with integer coefficients whose value is divisible by 3 for three integers k, k + 1, and k + 2, prove that f (m) is divisible by 3 for all integers m.55. (POL 4)IMO3 Find the conditions on the positive real number a such that there exists a tetrahedron k of whose edges (k = 1, 2, 3, 4, 5) have length a, and the other 6 − k edges have length 1.56. (POL 5) Let a and b be two natural numbers that have an equal number n of digits in their decimal expansions. The first m digits (from left to right) of the numbers a and b are equal. Prove that if m > n/2, then 1 a1/n − b1/n < . n57. (POL 6) On the sides AB and AC of triangle ABC two points K and L are given such that KB + LC = 1. Prove that KL passes through the AK AL centroid of ABC.58. (SWE 1) Six points P1 , . . . , P6 are given in 3-dimensional space such that no four of them lie in the same plane. Each of the line segments Pj Pk is colored black or white. Prove that there exists one triangle Pj Pk Pl whose edges are of the same color.59. (SWE 2) For each λ (0 < λ < 1 and λ = 1/n for all n = 1, 2, 3, . . . ) construct a continuous function f such that there do not exist x, y with 0 < λ < y = x + λ ≤ 1 for which f (x) = f (y).60. (SWE 3) Find the natural number n with the following properties: (1) Let S = {p1 , p2 , . . . } be an arbitrary finite set of points in the plane, and rj the distance from Pj to the origin O. We assign to each Pj the closed disk Dj with center Pj and radius rj . Then some n of these disks contain all points of S. (2) n is the smallest integer with the above property.61. (SWE 4) Let a0 , a1 , a2 be determined with a0 = 0, an+1 = 2an + 2n . Prove that if n is power of 2, then so is an .62. (SWE 5) Which natural numbers can be expressed as the difference of squares of two integers?63. (SWE 6) Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.64. (USS 1) Prove that for a natural number n > 2, (n!)! > n[(n − 1)!]n! .65. (USS 2) Prove that for a > b2 , √ 3 1 a − b a + b a − b a + ··· = a − b2 − b. 4 2
64 3 Problems3.12 The Twelfth IMOBudapest–Keszthely, Hungary, July 8–22, 19703.12.1 Contest Problems First Day (July 13)1. Given a point M on the side AB of the triangle ABC, let r1 and r2 be the radii of the inscribed circles of the triangles ACM and BCM respectively while ρ1 and ρ2 are the radii of the excircles of the triangles ACM and BCM at the sides AM and BM respectively. Let r and ρ denote the respective radii of the inscribed circle and the excircle at the side AB of the triangle ABC. Prove that r1 r2 r = . ρ1 ρ2 ρ2. Let a and b be the bases of two number systems and let An = x1 x2 . . . xn (a) , An+1 = x0 x1 x2 . . . xn (a) , Bn = x1 x2 . . . xn (b) Let 1 = a0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ · · · be a sequence of real numbers. Consider the sequence b1 , b2 , . . . defined by n ak−1 1 bn = 1− √ . ak ak k=1 Prove that: (a) For all natural numbers n, 0 ≤ bn < 2. (b) Given an arbitrary 0 ≤ b < 2, there is a sequence a0 , a1 , . . . , an , . . . of the above type such that bn > b is true for an infinity of natural numbers n. Second Day (July 14)4. For what natural numbers n can the product of some of the numbers n, n + 1, n + 2, n + 3, n + 4, n + 5 be equal to the product of the remaining ones?
3.12 IMO 1970 6717. (BUL 6) (SL70-3). Original formulation. In a triangular pyramid SABC one of the angles at S is right and the projection of S onto the base ABC is the orthocenter of ABC. Let r be the radius of the circle inscribed in the base, SA = m, SB = n, SC = p, H the height of the pyramid (through S), and r1 , r2 , r3 the radii of the circles inscribed in the intersections of the pyramid with the planes determined by the altitude of the pyramid and the lines SA, SB, SC respectively. Prove that: (a) m2 + n2 + p2 ≥ 18r2 ; (b) the ratios r1 /H, r2 /H, r3 /H lie in the interval [0.4, 0.5].18. (CZS 1) (SL70-4).19. (CZS 2) Let n > 1 be a natural number, a ≥ 1 a real number, and xk+1 x1 , x2 , . . . , xn numbers such that x1 = 1, xk = a+αk for k = 1, 2, . . . , n− 1, where αk are real numbers with αk ≤ k(k+1) . Prove that 1 √ 1 n−1 xn < a + . n−120. (CZS 3) (SL70-5).21. (CZS 4) Find necessary and sufficient conditions on given positive num- bers u, v for the following claim to be valid: there exists a right-angled triangle ABC with CD = u, CE = v, where D, E are points of the segments AB such that AD = DE = EB = 1 AB. 322. (FRA 1) (SL70-6).23. (FRA 2) Let E be a finite set, PE the family of its subsets, and f a mapping from PE to the set of nonnegative real numbers such that for any two disjoint subsets A, B of E, f (A ∪ B) = f (A) + f (B). Prove that there exists a subset F of E such that if with each A ⊂ E we associate a subset A consisting of elements of A that are not in F , then f (A) = f (A ), and f (A) is zero if and only if A is a subset of F .24. (FRA 3) Let n and p be two integers such that 2p ≤ n. Prove the inequality n−2p (n − p)! n+1 ≤ . p! 2 For which values does equality hold?25. (FRA 4) Suppose that f is a real function defined for 0 ≤ x ≤ 1 having the first derivative f for 0 ≤ x ≤ 1 and the second derivative f for 0 < x < 1. Prove that if f (0) = f (0) = f (1) = f (1) − 1 = 0,
68 3 Problems there exists a number 0 < y < 1 such that |f (y)| ≥ 4.26. (FRA 5) Consider a finite set of vectors in space {a1 , a2 , . . . , an } and the set E of all vectors of the form x = λ1 a1 + λ2 a2 + · · · + λn an , where λi are nonnegative numbers. Let F be the set consisting of all the vectors in E and vectors parallel to a given plane P . Prove that there exists a set of vectors {b1 , b2 , . . . , bp } such that F is the set of all vectors y of the form y = µ1 b1 + µ2 b2 + · · · + µp bp , where the µj are nonnegative.27. (FRA 6) Find a natural number n such that for all prime numbers p, n is divisible by p if and only if n is divisible by p − 1.28. (GDR 1) A set G with elements u, v, w, . . . is a group if the following conditions are fulfilled: (1) There is a binary algebraic operation ◦ defined on G such that for all u, v ∈ G there is a w ∈ G with u ◦ v = w. (2) This operation is associative; i.e., for all u, v, w ∈ G, (u ◦ v) ◦ w = u ◦ (v ◦ w). (3) For any two elements u, v ∈ G there exists an element x ∈ G such that u ◦ x = v, and an element y ∈ G such that y ◦ u = v. Let K be a set of all real numbers greater than 1. On K is defined an operation by a ◦ b = ab + (a2 − 1)(b2 − 1). Prove that K is a group.29. (GDR 2) Prove that the equation 4x +6x = 9x has no rational solutions.30. (GDR 3) (SL70-9).31. (GDR 4) Prove that for any triangle with sides a, b, c and area P the following inequality holds: √ 3 P ≤ (abc)2/3 . 4 Find all triangles for which equality holds.32. (NET 1) Let there be given an acute angle ∠AOB = 3α, where OA = OB. The point A is the center of a circle with radius OA. A line s parallel to OA passes through B. Inside the given angle a variable line t is drawn through O. It meets the circle in O and C and the given line s in D, where ∠AOC = x. Starting from an arbitrarily chosen position t0 of t, the series t0 , t1 , t2 , . . . is determined by defining BDi+1 = OCi for each i (in which Ci and Di denote the positions of C and D, corresponding to ti ). Making use of the graphical representations of BD and OC as functions of x, determine the behavior of ti for i → ∞.
3.12 IMO 1970 6933. (NET 2) The vertices of a given square are clockwise lettered A, B, C, D. On the side AB is situated a point E such that AE = AB/3. Starting from an arbitrarily chosen point P0 on segment AE and go- ing clockwise around the perimeter of the square, a series of points P0 , P1 , P2 , . . . is marked on the perimeter such that Pi Pi+1 = AB/3 for each i. It will be clear that when P0 is chosen in A or in E, then some Pi will coincide with P0 . Does this possibly also happen if P0 is chosen otherwise?34. (NET 3) In connection with a convex pentagon ABCDE we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.35. (NET 4) Find for every value of n a set of numbers p for which the fol- lowing statement is true: Any convex n-gon can be divided into p isosceles triangles. Alternative version. The same about division into p polygons with axis of symmetry.36. (NET 5) Let x, y, z be nonnegative real numbers satisfying x2 + y 2 + z 2 = 5 and yz + zx + xy = 2. Which values can the greatest of the numbers x2 − yz, y 2 − xz, z 2 − xy have?37. (NET 6) Solve the set of simultaneous equations v 2 + w2 + x2 + y 2 = 6 − 2u, 2 u + w2 + x2 + y 2 = 6 − 2v, u2 + 2 v + x2 + y 2 = 6 − 2w, u2 + 2 v +w + 2 y2 = 6 − 2x, u2 + 2 2 v +w + x 2 = 6 − 2y.38. (POL 1) Find the greatest integer A for which in any permutation of the numbers 1, . . . , 100 there exist ten consecutive numbers whose sum is at least A.39. (POL 2) (SL70-8).40. (POL 5) Let ABC be a triangle with angles α, β, γ commensurable with π. Starting from a point P interior to the triangle, a ball reflects on the sides of ABC, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices A, B, C, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment 0 to infinity consists of segments parallel to a finite set of lines.
70 3 Problems41. (POL 6) Let a cube of side 1 be given. Prove that there exists a point A on the surface S of the cube such that every point of S can be joined to A by a path on S of length not exceeding 2. Also prove that there is a point of S that cannot be joined with A by a path on S of length less than 2.42. (ROM 1) (SL70-2).43. (ROM 2) Prove that the equation π 2 π x3 − 3 tan x − 3x + tan =0 12 12 π has one root x1 = tan 36 , and find the other roots.44. (ROM 3) If a, b, c are side lengths of a triangle, prove that (a + b)(b + c)(c + a) ≥ 8(a + b − c)(b + c − a)(c + a − b).45. (ROM 4) Let M be an interior point of tetrahedron V ABC. Denote by A1 , B1 , C1 the points of intersection of lines M A, M B, M C with the planes V BC, V CA, V AB, and by A2 , B2 , C2 the points of intersection of lines V A1 , V B1 , V C1 with the sides BC, CA, AB. (a) Prove that the volume of the tetrahedron V A2 B2 C2 does not exceed one-fourth of the volume of V ABC. (b) Calculate the volume of the tetrahedron V1 A1 B1 C1 as a function of the volume of V ABC, where V1 is the point of intersection of the line V M with the plane ABC, and M is the barycenter of V ABC.46. (ROM 5) Given a triangle ABC and a plane π having no common points with the triangle, find a point M such that the triangle determined by the points of intersection of the lines M A, M B, M C with π is congruent to the triangle ABC.47. (ROM 6) Given a polynomial P (x) = ab(a − c)x3 + (a3 − a2 c + 2ab2 − b2 c + abc)x2 +(2a2 b + b2 c + a2 c + b3 − abc)x + ab(b + c), where a, b, c = 0, prove that P (x) is divisible by Q(x) = abx2 + (a2 + b2 )x + ab and conclude that P (x0 ) is divisible by (a + b)3 for x0 = (a + b + 1)n , n ∈ N.48. (ROM 7) Let a polynomial p(x) with integer coefficients take the value 5 for five different integer values of x. Prove that p(x) does not take the value 8 for any integer x.
3.12 IMO 1970 7149. (SWE 1) For n ∈ N, let f (n) be the number of positive integers k ≤ n that do not contain the digit 9. Does there exist a positive real number p such that f (n) ≥ p for all positive integers n? n50. (SWE 2) The area of a triangle is S and the sum of the lengths of its √ sides is L. Prove that 36S ≤ L2 3 and give a necessary and sufficient condition for equality.51. (SWE 3) Let p be a prime number. A rational number x, with 0 < x < 1, is written in lowest terms. The rational number obtained from x by adding p to both the numerator and the denominator differs from x by 1/p2 . Determine all rational numbers x with this property.52. (SWE 4) (SL70-10).53. (SWE 5) A square ABCD is divided into (n − 1)2 congruent squares, with sides parallel to the sides of the given square. Consider the grid of all n2 corners obtained in this manner. Determine all integers n for which it is possible to construct a nondegenerate parabola with its axis parallel to one side of the square and that passes through exactly n points of the grid.54. (SWE 6) (SL70-11).55. (USS 1) A turtle runs away from an UFO with a speed of 0.2 m/s. The UFO flies 5 meters above the ground, with a speed of 20 m/s. The UFO's path is a broken line, where after flying in a straight path of length (in meters) it may turn through for any acute angle α such that tan α < 1000 . When the UFO's center approaches within 13 meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.56. (USS 2) A square hole of depth h whose base is of length a is given. A dog is tied to the center of the square at the bottom of the hole by a √ rope of length L > 2a2 + h2 , and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).57. (USS 3) Let the numbers 1, 2, . . . , n2 be written in the cells of an n × n square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the kth row? (k a positive integer, 1 ≤ k ≤ n.)58. (USS 4) (SL70-12).59. (USS 5) (SL70-7).
72 3 Problems3.12.3 Shortlisted Problems1. (BEL 3) Consider a regular 2n-gon and the n diagonals of it that pass through its center. Let P be a point of the inscribed circle and let a1 , a2 , . . . , an be the angles in which the diagonals mentioned are visible from the point P . Prove that n π cos2 tan2 ai = 2n 2n . i=1 sin4 2n π2. (ROM 1)IMO2 Let a and b be the bases of two number systems and let An = x1 x2 . . . xn (a) , An+1 = x0 x1 x2 . . . xn (a) , (b) Bn = x1 x2 . . . xn (BUL 6)IMO5 In the tetrahedron SABC the angle BSC is a right angle, and the projection of the vertex S to the plane ABC is the intersection of the altitudes of the triangle ABC. Let z be the radius of the inscribed circle of the triangle ABC. Prove that SA2 + SB 2 + SC 2 ≥ 18z 2 .4. (CZS 1)IMO4 For what natural numbers n can the product of some of the numbers n, n + 1, n + 2, n + 3, n + 4, n + 5 be equal to the product of the remaining ones?5. (CZS 3) Let M be an interior point of the tetrahedron ABCD. Prove that −→ − −→ − M A vol(M BCD) + M B vol(M ACD) −→ − −→− +M C vol(M ABD) + M D vol(M ABC) = 0 (vol(P QRS) denotes the volume of the tetrahedron P QRS).6. (FRA 1) In the triangle ABC let B and C be the midpoints of the sides AC and AB respectively and H the foot of the altitude passing through the vertex A. Prove that the circumcircles of the triangles AB C , BC H, and B CH have a common point I and that the line HI passes through the midpoint of the segment B C .
76 3 Problems 9. (BUL 6) The base of an inclined prism is a triangle ABC. The per- pendicular projection of B1 , one of the top vertices, is the midpoint of BC. The dihedral angle between the lateral faces through BC and AB is α, and the lateral edges of the prism make an angle β with the base. If r1 , r2 , r3 are exradii of a perpendicular section of the prism, assuming that in ABC, cos2 A + cos2 B + cos2 C = 1, ∠A < ∠B < ∠C, and BC = a, calculate r1 r2 + r1 r3 + r2 r3 .10. (CUB 1) In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?11. (CUB 2) Prove that n! cannot be the square of any natural number.12. (CUB 3) A system of n numbers x1 , x2 , . . . , xn is given such that x1 = logxn−1 xn , x2 = logxn x1 , ... , xn = logxn−2 xn−1 . n Prove that k=1 xk = 1.13. (CUB 4) One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: Martian : I have spent 1/12 of my life on Pluton. Human : I also have. Venusian : Me too. Martian : But Venusian and I have spend much more time here than you, Human. Human : That is true. However, Venusian and I are of the same age. Venusian : Yes, I have lived 300 Earth years. Martian : Venusian and I have been on Pluton for the past 13 years. It is known that Human and Martian together have lived 104 Earth years. Find the ages of Martian, Venusian, and Human.514. (GBR 1) Note that 83 − 73 = 169 = 132 and 13 = 22 + 32 . Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.15. (GBR 2) Let ABCD be a convex quadrilateral whose diagonals intersect at O at an angle θ. Let us set OA = a, OB = b, OC = c, and OD = d, c > a > 0, and d > b > 0. Show that if there exists a right circular cone with vertex V , with the properties: (1) its axis passes through O, and (2) its curved surface passes through A, B, C and D, then d2 b2 (c + a)2 − c2 a2 (d + b)2 OV 2 = . ca(d − b)2 − db(c − a)2 5 The numbers in the problem are not necessarily in base 10.
3.13 IMO 1971 77 Show also that if c+a lies between ca and ca , and c−a = ca , then for d+b db db d−b db a suitable choice of θ, a right circular cone exists with properties (1) and (2).16. (GBR 3) (SL71-4). Original formulation. Two (intersecting) circles are given and a point P through which it is possible to draw a straight line on which the circles intercept two equal chords. Describe a construction by straightedge and compass for the straight line and prove the validity of your construction.17. (GDR 1) (SL71-3). Original formulation. Find all solutions of the system x + y + z = 3, x + y 3 + z 3 = 15, 3 x5 + y 5 + z 5 = 83.18. (GDR 2) Let a1 , a2 , . . . , an be positive numbers, mg = (a1 a2 · · · an )1/n their geometric mean, and ma = (a1 + a2 + · · · + an )/n their arithmetic mean. Prove that (1 + mg )n ≤ (1 + a1 ) · · · (1 + an ) ≤ (1 + ma )n .19. (GDR 3) In a triangle P1 P2 P3 let Pi Qi be the altitude from Pi for i = 1, 2, 3 (Qi being the foot of the altitude). The circle with diameter Pi Qi meets the two corresponding sides at two points different from Pi . Denote the length of the segment whose endpoints are these two points by li . Prove that l1 = l2 = l3 .20. (GDR 4) Let M be the circumcenter of a triangle ABC. The line through M perpendicular to CM meets the lines CA and CB at Q and P respec- tively. Prove that CP CQ AB = 2. CM CM P Q21. (HUN 1) (SL71-5).22. (HUN 2) We are given an n × n board, where n is an odd number. In each cell of the board either +1 or −1 is written. Let ak and bk denote the products of numbers in the kth row and in the kth column respectively. Prove that the sum a1 + a2 + · · · + an + b1 + b2 + · · · + bn cannot be equal to zero.23. (HUN 3) Find all integer solutions of the equation x2 + y 2 = (x − y)3 .24. (HUN 4) Let A, B, and C denote the angles of a triangle. If sin2 A + sin2 B + sin2 C = 2, prove that the triangle is right-angled.
78 3 Problems25. (HUN 5) Let ABC, AA1 A2 , BB1 B2 , CC1 C2 be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments A2 B1 , B2 C1 , C2 A1 by P, Q, R in this order. Prove that the triangle P QR is equilateral.26. (HUN 6) An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates (0, 0), (p, 0), (p, q), (0, q) for some positive integers p, q. Show that there must exist two among them one of which is entirely contained in the other.27. (HUN 7) (SL71-6).28. (NET 1) (SL71-7). Original formulation. A tetrahedron ABCD is given. The sum of angles of the tetrahedron at the vertex A (namely ∠BAC, ∠CAD, ∠DAB) is de- noted by α, and β, γ, δ are defined analogously. Let P, Q, R, S be variable points on edges of the tetrahedron: P on AD, Q on BD, R on BC, and S on AC, none of them at some vertex of ABCD. Prove that: (a) if α + β = 2π, then P Q + QR + RS + SP attains no minimal value; (b) if α + β = 2π, then α γ α AB sin = CD sin and P Q + QR + RS + SP ≥ 2AB sin . 2 2 229. (NET 2) A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii r1 , r2 , while the incircle has radius r. Given that r1 and r2 are natural numbers and that r1 r2 = r, find r1 , r2 , and r.30. (NET 3) Prove that the system of equations 2yz + x − y − z = a, 2xz − x + y − z = a, 2xy − x − y + z = a, a being a parameter, cannot have five distinct solutions. For what values of a does this system have four distinct integer solutions?31. (NET 4) (SL71-8).32. (NET 5) Two half-lines a and b, with the common endpoint O, make an acute angle α. Let A on a and B on b be points such that OA = OB, and let b be the line through A parallel to b. Let β be the circle with center B and radius BO. We construct a sequence of half-lines c1 , c2 , c3 , . . . , all lying inside the angle α, in the following manner: (i) c1 is given arbitrarily;
3.13 IMO 1971 79 (ii) for every natural number k, the circle β intercepts on ck a segment that is of the same length as the segment cut on b by a and ck+1 . Prove that the angle determined by the lines ck and b has a limit as k tends to infinity and find that limit.33. (NET 6) A square 2n × 2n grid is given. Let us consider all possible paths along grid lines, going from the center of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its center at the grid center is passed through only once. (a) Prove that the number of all such paths is equal to 4 n (16i − 9). i=2 (b) Find the number of pairs of such paths that divide the grid into two congruent figures. (c) How many quadruples of such paths are there that divide the grid into four congruent parts?34. (POL 1) (SL71-9).35. (POL 2) (SL71-10).36. (POL 3) (SL71-11).37. (POL 4) Let S be a circle, and α = {A1 , . . . , An } a family of open arcs in S. Let N (α) = n denote the number of elements in α. We say that α n is a covering of S if k=1 Ak ⊃ S. Let α = {A1 , . . . , An } and β = {B1 , . . . , Bm } be two coverings of S. Show that we can choose from the family of all sets Ai ∩ Bj , i = 1, 2, . . . , n, j = 1, 2, . . . , m, a covering γ of S such that N (γ) ≤ N (α) + N (β).38. (POL 5) Let A, B, C be three points with integer coordinates in the plane and K a circle with radius R passing through A, B, C. Show that AB·BC·CA ≥ 2R, and if the center of K is in the origin of the coordinates, show that AB · BC · CA ≥ 4R.39. (POL 6) (SL71-12).40. (SWE 1) Prove that 1 1 1 1 1 1− 1− 1− ··· 1 − > , n = 2, 3, . . . . 23 33 43 n3 241. (SWE 2) Consider the set of grid points (m, n) in the plane, m, n inte- gers. Let σ be a finite subset and define S(σ) = (100 − |m| − |n|). (m,n)∈σ Find the maximum of S, taken over the set of all such subsets σ.42. (SWE 3) Let Li , i = 1, 2, 3, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths li , i = 1, 2, 3. By L∗ we i
80 3 Problems denote the segment of length li with its midpoint on the midpoint of the corresponding side of the triangle. Let M (L) be the set of points in the plane whose orthogonal projections on the sides of the triangle are in L1 , L2 , and L3 , respectively; M (L∗ ) is defined correspondingly. Prove that if l1 ≥ l2 + l3 , we have that the area of M (L) is less than or equal to the area of M (L∗ ).43. (SWE 4) Show that for nonnegative real numbers a, b and integers n ≥ 2, n an + b n a+b ≥ . 2 2 When does equality hold?44. (SWE 5) (SL71-13).45. (SWE 6) Let m and n denote integers greater than 1, and let ν(n) be the number of primes less than or equal to n. Show that if the equation ν(n) = m has a solution, then so does the equation ν(n) = m − 1. n n46. (USS 1) (SL71-14).47. (USS 2) (SL71-15).48. (USS 3) A sequence of real numbers x1 , x2 , . . . , xn is given such that xi+1 = xi + 30000 1 − x2 , i = 1, 2, . . . , and x1 = 0. Can n be equal to 1 i 50000 if xn < 1?49. (USS 4) Diagonals of a convex quadrilateral ABCD intersect at a point O. Find all angles of this quadrilateral if OBA = 30◦ , OCB = 45◦ , ODC = 45◦ , and OAD = 30◦ .50. (USS 5) (SL71-16).51. (USS 6) Suppose that the sides AB and DC of a convex quadrilateral ABCD are not parallel. On the sides BC and AD, pairs of points (M, N ) and (K, L) are chosen such that BM = M N = N C and AK = KL = LD. Prove that the areas of triangles OKM and OLN are different, where O is the intersection point of AB and CD.52. (YUG 1) (SL71-17).53. (YUG 2) Denote by xn (p) the multiplicity of the prime p in the canonical (p) representation of the number n! as a product of primes. Prove that xnn < 1 xn (p) 1 p−1 and limn→∞ n = p−1 .54. (YUG 3) A set M is formed of 2n men, n = 1, 2, . . . . Prove that we n can choose a subset P of the set M consisting of n + 1 men such that one of the following conditions is satisfied: (1) every member of the set P knows every other member of the set P ; (2) no member of the set P knows any other member of the set P .
3.13 IMO 1971 8313. (SWE 5)IMO6 Consider the n × n array of nonnegative integers ⎛ ⎞ a11 a12 . . . a1n ⎜ a21 a22 · · · a2n ⎟ ⎜ ⎟ ⎜ . . . ⎟, ⎝ . . . . . ⎠ . an1 an2 . . . ann with the following property: If an element aij is zero, then the sum of the elements of the ith row and the jth column is greater than or equal to n. Prove that the sum of all the elements is greater than or equal to 1 n2 . 214. (USS 1) A broken line A1 A2 . . . An is drawn in a 50 × 50 square, so that the distance from any point of the square to the broken line is less than 1. Prove that its total length is greater than 1248.15. (USS 2) Natural numbers from 1 to 99 (not necessarily distinct) are written on 99 cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by 100. Show that all the cards contain the same number.16. (USS 5)IMO2 Given a convex polyhedron P1 with 9 vertices A1 , . . . , A9 , let us denote by P2 , P3 , . . . , P9 the images of P1 under the translations mapping the vertex A1 to A2 , A3 , . . . , A9 respectively. Prove that among the polyhedra P1 , . . . , P9 at least two have a common interior point.17. (YUG 1) Prove the inequality a1 + a3 a2 + a4 a3 + a1 a4 + a2 + + + ≥ 4, a1 + a2 a2 + a3 a3 + a4 a4 + a1 where ai > 0, i = 1, 2, 3, 4.
3.14 IMO 1972 85 x4 = yz − x2 + a, y 4 = zx − y 2 + a, z 4 = xy − z 2 + a, has at most one real solution. 3. (BUL 3) On a line a set of segments is given of total length less than n. Prove that every set of n points of the line can be translated in some direction along the line for a distance smaller than n/2 so that none of the points remain on the segments. 4. (BUL 4) Given a triangle, prove that the points of intersection of three pairs of trisectors of the inner angles at the sides lying closest to those sides are vertices of an equilateral triangle. 5. (BUL 5) Given a pyramid whose base is an n-gon inscribable in a circle, let H be the projection of the top vertex of the pyramid to its base. Prove that the projections of H to the lateral edges of the pyramid lie on a circle. 6. (BUL 6) Prove the inequality π π (n + 1) cos − n cos > 1 n+1 n for all natural numbers n ≥ 2. 7. (BUL 7) (SL72-1). 8. (CZS 1) (SL72-2). 9. (CZS 2) Given natural numbers k and n, k ≤ n, n ≥ 3, find the set of all values in the interval (0, π) that the kth-largest among the interior angles of a convex ngon can take.10. (CZS 3) Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.11. (CZS 4) (SL72-3).12. (CZS 5) A circle k = (S, r) is given and a hexagon AA BB CC inscribed in it. The lengths of sides of the hexagon satisfy AA = A B, BB = B C, CC = C A. Prove that the area P of triangle ABC is not greater than the area P of triangle A B C . When does P = P hold?13. (CZS 6) Given a sphere K, determine the set of all points A that are vertices of some parallelograms ABCD that satisfy AC ≤ BD and whose entire diagonal BD is contained in K.14. (GBR 1) (SL72-7).15. (GBR 2) (SL72-8).
86 3 Problems16. (GBR 3) Consider the set S of all the different odd positive integers that are not multiples of 5 and that are less than 30m, m being a positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two integers one of which divides the other? Prove your result.17. (GBR 4) A solid right circular cylinder with height h and base-radius r has a solid hemisphere of radius r resting upon it. The center of the hemisphere O is on the axis of the cylinder. Let P be any point on the surface of the hemisphere and Q the point on the base circle of the cylinder that is furthest from P (measuring along the surface of the combined solid). A string is stretched over the surface from P to Q so as to be as short as possible. Show that if the string is not in a plane, the straight line P O when produced cuts the curved surface of the cylinder.18. (GBR 5) We have p players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers s1 ≤ s2 ≤ s3 ≤ · · · ≤ sp is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if p k 1 1 (i) si = p(p − 1) and (ii) for all k < p, si ≥ k(k − 1). i=1 2 i=1 219. (GBR 6) Let S be a subset of the real numbers with the following properties: (i) If x ∈ S and y ∈ S, then x − y ∈ S; (ii) If x ∈ S and y ∈ S, then xy ∈ S; (iii) S contains an exceptional number x such that there is no number y in S satisfying x y + x + y = 0; (iv) If x ∈ S and x = x , there is a number y in S such that xy + x + y = 0. Show that (a) S has more than one number in it; (b) x = −1 leads to a contradiction; (c) x ∈ S and x = 0 implies 1/x ∈ S.20. (GDR 1) (SL72-4).21. (GDR 2) (SL72-5).22. (GDR 3) (SL72-6).23. (MON 1) Does there exist a 2n-digit number a2n a2n−1 . . . a1 (for an arbitrary n) for which the following equality holds: a2n . . . a1 = (an . . . a1 )2 ?24. (MON 2) The diagonals of a convex 18-gon are colored in 5 different colors, each color appearing on an equal number of diagonals. The diag- onals of one color are numbered 1, 2, . . . . One randomly chooses one-fifth
3.14 IMO 1972 87 of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly n pairs of diagonals of the same color and with fixed indices i, j.25. (NET 1) We consider n real variables xi (1 ≤ i ≤ n), where n is an integer and n ≥ 2. The product of these variables will be denoted by p, their sum by s, and the sum of their squares by S. Furthermore, let α be a positive constant. We now study the inequality ps ≤ S α . Prove that it holds for every n-tuple (xi ) if and only if α = n+1 . 226. (NET 2) (SL72-9).27. (NET 3) (SL72-10).28. (NET 4) The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.29. (NET 5) Let A, B, C be points on the sides B1 C1 , C1 A1 , A1 B1 of a triangle A1 B1 C1 such that A1 A, B1 B, C1 C are the bisectors of angles of the triangle. We have that AC = BC and A1 C1 = B1 C1 . (a) Prove that C1 lies on the circumcircle of the triangle ABC. (b) Suppose that BAC1 = π/6; find the form of triangle ABC.30. (NET 6) (SL72-11). 3n −231. (ROM 1) Find values of n ∈ N for which the fraction 2n −3 is reducible.32. (ROM 2) If n1 , n2 , . . . , nk are natural numbers and n1 +n2 +· · ·+nk = n, show that max n1 n2 · · · nk = (t + 1)r tk−r , n1 +···+nk =n where t = [n/k] and r is the remainder of n upon division by k; i.e., n = tk + r, 0 ≤ r ≤ k − 1.33. (ROM 3) A rectangle ABCD is given whose sides have lengths 3 and 2n, where n is a natural number. Denote by U (n) the number of ways in which one can cut the rectangle into rectangles of side lengths 1 and 2. (a) Prove that U (n + 1) + U √ − 1) = 4U√ (n (n); √ √ (b) Prove that U (n) = 2√3 [( 3 + 1)(2 + 3)n + ( 3 − 1)(2 − 3)n ]. 134. (ROM 4) If p is a prime number greater than 2 and a, b, c integers not divisible by p, prove that the equation ax2 + by 2 = pz + c has an integer solution.35. (ROM 5) (a) Prove that for a, b, c, d ∈ R, m ∈ [1, +∞) with am + b = −cm + d = m, √ √ 4m2 (i) a2 + b2 + c2 + d2 + (a − c)2 + (b − d)2 ≥ 1+m2 , and
88 3 Problems 2 (ii) 2 ≤ 1+m2 < 4. 4m (b) Express a, b, c, d as functions of m so that there is equality in (1).36. (ROM 6) A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.37. (SWE 1) On a chessboard (8 × 8 squares with sides of length 1) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths 1 and 2?38. (SWE 2) Congruent rectangles with sides m (cm) and n (cm) are given (m, n positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)39. (SWE 3) How many tangents to the curve y = x3 − 3x (y = x3 + px) can be drawn from different points in the plane?40. (SWE 4) Prove the inequalities u sin u πu π ≤ ≤ , for 0 ≤ u < v ≤ . v sin v 2v 241. (SWE 5) The ternary expansion x = 0.10101010 . . . is given. Give the binary expansion of x. Alternatively, transform the binary expansion y = 0.110110110 . . . into a ternary expansion.42. (SWE 6) The decimal number 13101 is given. It is instead written as a ternary number. What are the two last digits of this ternary number?43. (USS 1) A fixed point A inside a circle is given. Consider all chords XY of the circle such that ∠XAY is a right angle, and for all such chords construct the point M symmetric to A with respect to XY . Find the locus of points M .44. (USS 2) (SL72-12).45. (USS 3) Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect at point O. Let a line through O intersect segment AB at M and segment CD at N . Prove that the segment M N is not longer than at least one of the segments AC and BD.46. (USS 4) Numbers 1, 2, . . . , 16 are written in a 4×4 square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers 1 and 16 lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals 17.
3.14 IMO 1972 893.14.3 Shortlisted Problems1. (BUL 7)IMO5 Let f and ϕ be real functions defined on the set R satisfying the functional equation f (x + y) + f (x − y) = 2ϕ(y)f (x), (1) for arbitrary real x, y (give examples of such functions). Prove that if f (x) is not identically 0 and |f (x)| ≤ 1 for all x, then |ϕ(x)| ≤ 1 for all x.2. (CZS 1) We are given 3n points A1 , A2 , . . . , A3n in the plane, no three of them collinear. Prove that one can construct n disjoint triangles with vertices at the points Ai .3. (CZS 4) Let x1 , x2 , . . . , xn be real numbers satisfying x1 +x2 +· · ·+xn = 0. Let m be the least and M the greatest among them. Prove that x2 + x2 + · · · + x2 ≤ −nmM. 1 2 n4. (GDR 1) Let n1 , n2 be positive integers. Consider in a plane E two dis- joint sets of points M1 and M2 consisting of 2n1 and 2n2 points, respec- tively, and such that no three points of the union M1 ∪ M2 are collinear. Prove that there exists a straightline g with the following property: Each of the two half-planes determined by g on E (g not being included in either) contains exactly half of the points of M1 and exactly half of the points of M2 .5. (GDR 2) Prove the following assertion: The four altitudes of a tetrahe- dron ABCD intersect in a point if and only if AB 2 + CD2 = BC 2 + AD2 = CA2 + BD2 .6. (GDR 3) Show that for any n ≡ 0 (mod 10) there exists a multiple of n not containing the digit 0 in its decimal expansion.7. (GBR 1)IMO6 (a) A plane π passes through the vertex O of the regular tetrahedron OP QR. We define p, q, r to be the signed distances of P, Q, R from π measured along a directed normal to π. Prove that p2 + q 2 + r2 + (q − r)2 + (r − p)2 + (p − q)2 = 2a2 , where a is the length of an edge of a tetrahedron. (b) Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.8. (GBR 2)IMO3 Let m and n be nonnegative integers. Prove that m!n!(m+ n)! divides (2m)!(2n)!.9. (NET 2)IMO4 Find all solutions in positive real numbers xi (i = 1, 2, 3, 4, 5) of the following system of inequalities:
3.15 IMO 1973 913.15 The Fifteenth IMOMoscow, Soviet Union, July 5–16, 19733.15.1 Contest Problems First Day (July 9) −→ −→ − − −→ −1. Let O be a point on the line l and OP1 , OP2 , . . . , OPn unit vectors such that points P1 , P2 , . . . , Pn and line l lie in the same plane and all points Pi lie in the same half-plane determined by l. Prove that if n is odd, then −→ −→ − − −→ − OP1 + OP2 + · · · + OPn ≥ 1. −→ − −→ − ( OM is the length of vector OM ).2. but not equal?3. Determine the minimum of a2 + b2 if a and b are real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. Second Day (July 10)4. A soldier has to investigate whether there are mines in an area that has the form of equilateral triangle. The radius of his detector's range is equal to one-half the altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the smallest path through which the soldier has to pass in order to check the entire region.5. Let G be the set of functions f : R → R of the form f (x) = ax + b, where a and b are real numbers and a = 0. Suppose that G satisfies the following conditions: (1) If f, g ∈ G, then g ◦ f ∈ G, where (g ◦ f )(x) = g[f (x)]. (2) If f ∈ G and f (x) = ax + b, then the inverse f −1 of f belongs to G (f −1 (x) = (x − b)/a). (3) For each f ∈ G there exists a number xf ∈ R such that f (xf ) = xf . Prove that there exists a number k ∈ R such that f (k) = k for all f ∈ G.6. b (2) q < k
92 3 Problems3.15.2 Shortlisted Problems 1. (BUL 6) Let a tetrahedron ABCD be inscribed in a sphere S. Find the locus of points P inside the sphere S for which the equality AP BP CP DP + + + =4 P A1 P B1 P C1 P D1 holds, where A1 , B1 , C1 , and D1 are the intersection points of S with the lines AP, BP, CP, and DP , respectively. 2. (CZS 1) Given a circle K, find the locus of vertices A of parallelograms ABCD with diagonals AC ≤ BD, such that BD is inside K. 3. (CZS 6)IMO1 Prove that the sum of an odd number of unit vectors passing through the same point O and lying in the same half-plane whose border passes through O has length greater than or equal to 1. 4. (GBR 1) Let P be a set of 7 different prime numbers and C a set of 28 different composite numbers each of which is a product of two (not necessarily different) numbers from P . The set C is divided into 7 disjoint four-element subsets such that each of the numbers in one set has a com- mon prime divisor with at least two other numbers in that set. How many such partitions of C are there? 5. (FRA 2) A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles. 6. (POL 2)IMO2? 7. (POL 3) Given a tetrahedron ABCD, let x = AB · CD, y = AC · BD, and z = AD · BC. Prove that there exists a triangle with edges x, y, z. k 8. (ROM 1) Prove that there are exactly [k/2] arrays a1 , a2 , . . . , ak+1 of nonnegative integers such that a1 = 0 and |ai −ai+1 | = 1 for i = 1, 2, . . . , k. 9. (ROM 2) Let Ox, Oy, Oz be three rays, and G a point inside the trihe- dron Oxyz. Consider all planes passing through G and cutting Ox, Oy, Oz at points A, B, C, respectively. How is the plane to be placed in order to yield a tetrahedron OABC with minimal perimeter?10. (SWE 3)IMO6 (2) q < bk11. (SWE 4)IMO3 Determine the minimum of a2 + b2 if a and b are real numbers for which the equation
94 3 Problems3.16 The Sixteenth IMOErfurt–Berlin, DR Germany, July 4–17, 19743.16.1 Contest Problems First Day (July 8)1. Alice, Betty, and Carol took the same series of examinations2. Let ABC be a triangle. Prove that there exists a point D on the side AB such that CD is the geometric mean of AD and BD if and only if √ C sin A sin B ≤ sin . 23. Prove that there does not exist a natural number n for which the number n 2n + 1 3k 2 2k + 1 k=0 is divisible by 5. Second Day (July 9)4. Consider a partition of an 8×8 chessboard into p rectangles whose interiors are disjoint such that each rectangle contains5. If a, b, c, d are arbitrary positive real numbers, find all possible values of a b c d S= + + + . a+b+d a+b+c b+c+d a+c+d6. Let P (x) be a polynomial with integer coefficients. If n(P ) is the number of (distinct) integers k such that P 2 (k) = 1, prove that n(P )−deg(P ) ≤ 2, where deg(P ) denotes the degree of the polynomial P .
3.16 IMO 1974 953.16.2 Longlisted Problems1. (BUL 1) (SL74-11).2. (BUL 2) Let {un } be the Fibonacci sequence, i.e., u0 = 0, u1 = 1, un = un−1 + un−2 for n > 1. Prove that there exist infinitely many prime numbers p that divide up−1 .3. (BUL 3) Let ABCD be an arbitrary quadrilateral. Let squares ABB1 A2 , BCC1 B2 , CDD1 C2 , DAA1 D2 be constructed in the exterior of the quadrilateral. Furthermore, let AA1 P A2 and CC1 QC2 be parallelograms. For any arbitrary point P in the interior of ABCD, parallelograms RASC and RP T Q are constructed. Prove that these two parallelograms have two vertices in common.4. (BUL 4) Let Ka , Kb , Kc with centers Oa , Ob , Oc be the excircles of a triangle ABC, touching the interiors of the sides BC, CA, AB at points Ta , Tb , Tc respectively. Prove that the lines Oa Ta , Ob Tb , Oc Tc are concurrent in a point P for which P Oa = P Ob = P Oc = 2R holds, where R denotes the circumradius of ABC. Also prove that the circumcenter O of ABC is the midpoint of the segment P J, where J is the incenter of ABC.5. (BUL 5) A straight cone is given inside a rectangular parallelepiped B, with the apex at one of the vertices, say T , of the parallelepiped, and the base touching the three faces opposite to T . Its axis lies at the long diagonal through T . If V1 and V2 are the volumes of the cone and the parallelepiped respectively, prove that √ 3πV2 V1 ≤ . 276. (CUB 1) Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.7. (CUB 2) Let P be a prime number and n a natural number. Prove that the product 2n−1 1 p2 i N = n2 ((p − 1)i)! p pi i=1; 2 i is a natural number that is not divisible by p.8. (CUB 3) (SL74-9).9. (CZS 1) Solve the following system of linear equations with unknown x1 , . . . , xn (n ≥ 2) and parameters c1 , . . . , cn :
96 3 Problems 2x1 −x2 = c1 ; −x1 +2x2 −x3 = c2 ; −x2 +2x3 −x4 = c3 ; ... ... ... ... −xn−2 +2xn−1 −xn = cn−1 ; −xn−1 +2xn = cn .10. (CZS 2) A regular octagon P is given whose incircle k has diameter 1. About k is circumscribed a regular 16-gon, which is also inscribed in P , cutting from P eight isosceles triangles. To the octagon P , three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every 11-gon so obtained is said to be P . Prove the following statement: Given a finite set M of points lying in P such that every two points of this set have a distance not exceeding 1, one of the 11-gons P contains all of M .11. (CZS 3) Given a line p and a triangle in the plane, construct an equilateral triangle one of whose vertices lies on the line p, while the other two halve the perimeter of .12. (CZS 4) A circle K with radius r, a point D on K, and a convex angle with vertex S and rays a and b are given in the plane. Construct a parallelogram ABCD such that A and B lie on a and b respectively, SA + SB = r, and C lies on K.13. (FIN 1) Prove that 2147 − 1 is divisible by 343.14. (FIN 2) Let n and k be natural numbers and a1 , a2 , . . . , an positive real numbers satisfying a1 + a2 + · · · + an = 1. Prove that a−k + a−k + · · · + a−k ≥ nk+1 . 1 2 n15. (FIN 3) (SL74-10).16. (GBR 1) A pack of 2n cards contains n different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players A and B. A third person called the dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver. A is initially the receiver and takes the first card dealt. The player who first obtains a complete set of n different cards wins the game. What fraction of all possible arrangements of the pack lead to A winning? Prove the correctness of your answer.17. (GBR 2) Show that there exists a set S of 15 distinct circles on the surface of a sphere, all having the same radius and such that 5 touch exactly 5 others, 5 touch exactly 4 others, and 5 touch exactly 3 others.
3.16 IMO 1974 9718. (GBR 3) (SL74-5).19. (GBR 4) (Alternative to GBR 2) Prove that there exists, for n ≥ 4, a set S of 3n equal circles in spacethat can be partitioned into three subsets s5 , s4 , and s3 , each containing n circles, such that each circle in sr touches exactly r circles in S.20. (NET 1) For which natural numbers n do there exist n natural numbers n −2 ai (1 ≤ i ≤ n) such that i=1 ai = 1?21. (NET 2) Let M be a nonempty subset of Z+ such that for every element √ x in M , the numbers 4x and [ x] also belong to M . Prove that M = Z+ .22. (NET 3) (SL74-8).23. (POL 1) (SL74-2).24. (POL 2) (SL74-7).25. (POL 3) Let f : R → R be of the form f (x) = x + ε sin x, where 0 < |ε| ≤ 1. Define for any x ∈ R, xn = f ◦ · · · ◦ f (x). n times Show that for every x ∈ R there exists an integer k such that limn→∞ xn = kπ.26. (POL 4) Let g(k) be the number of partitions of a k-element set M , i.e., the number of families {A1 , A2 , . . . , As } of nonempty subsets of M such n that Ai ∩ Aj = ∅ for i = j and i=1 Ai = M . Prove that nn ≤ g(2n) ≤ (2n)2n for every n.27. (ROM 1) Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary points on C1 and C2 respectively, and Q the midpoint of segment P1 P2 . Find the locus of points Q as P1 and P2 go through all possible positions. Alternative version. Let C1 , C2 , C3 be three circles in the same plane. Find the locus of the centroid of triangle P1 P2 P3 as P1 , P2 , and P3 go through all possible positions on C1 , C2 , and C3 respectively.28. (ROM 2) Let M be a finite set and P = {M1 , M2 , . . . , Mk } a partition of M (i.e., k Mi = M , Mi = ∅, Mi ∩ Mj = ∅ for all i, j ∈ {1, 2, . . . , k}, i=1 i = j). We define the following elementary operation on P : Choose i, j ∈ {1, 2, . . . , k}, such that i = j and Mi has a elements and Mj has b elements such that a ≥ b. Then take b elements from Mi and place them into Mj , i.e., Mj becomes the union of itself unifies and a b-element subset of Mi , while the same subset is subtracted from Mi (if a = b, Mi is thus removed from the partition). Let a finite set M be given. Prove that the property "for every partition P of M there exists a sequence P = P1 , P2 , . . . , Pr such that Pi+1 is obtained
3.16 IMO 1974 99 if there are integers i and j with i ≥ 0 and j ≥ 0 such that the three elements are (i) nij , ni,j+1 , ni,j+2 , or (ii) nij , ni+1,j , ni+2,j , or (iii) ni+2,j , ni+1,j+1 , ni,j+2 . An elementary operation on a diagram is an operation by which three adjacent elements nij are changed into nij in such a way that |nij − nij | = 1. Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?37. (USA 1) Let a, b, and c denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side a and then propelled toward side b with direction defined by the angle θ. For what values of θ will the ball strike the sides b, c, a in that order?38. (USA 2) Consider the binomial coefficients n = k n! k!(n−k)! (k = 1, n n n 2, . . . , n−1). Determine all positive integers n for which 1 , 2 , . . . , n−1 are all even numbers.39. (USA 3) Let n be a positive integer, n ≥ 2, and consider the polynomial equation xn − xn−2 − x + 2 = 0. For each n, determine all complex numbers x that satisfy the equation and have modulus |x| = 1.40. (USA 4) (SL74-1).41. (USA 5) Through the circumcenter O of an arbitrary acute-angled trian- gle, chords A1 A2 , B1 B2 , C1 C2 are drawn parallel to the sides BC, CA, AB of the triangle respectively. If R is the radius of the circumcircle, prove that A1 O · OA2 + B1 O · OB2 + C1 O · OC2 = R2 .42. (USS 1) (SL74-12).43. (USS 2) An (n2 + n + 1) × (n2 + n + 1) matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed (n + 1)(n2 + n + 1).44. (USS 3) We are given n mass points of equal mass in space. We define a sequence of points O1 , O2 , O3 , . . . as follows: O1 is an arbitrary point (within the unit distance of at least one of the n points); O2 is the center of gravity of all the n given points that are inside the unit sphere centered at O1 ; O3 is the center of gravity of all of the n given points that are inside the unit sphere centered at O2 ; etc. Prove that starting from some m, all points Om , Om+1 , Om+2 , . . . coincide.45. (USS 4) (SL74-4).
100 3 Problems46. (USS 5) Outside an arbitrary triangle ABC, triangles ADB and BCE are constructed such that ∠ADB = ∠BEC = 90◦ and ∠DAB = ∠EBC = 30◦ . On the segment AC the point F with AF = 3F C is chosen. Prove that ∠DF E = 90◦ and ∠F DE = 30◦ .47. (VIE 1) Given two points A, B outside of a given plane P , find the MA positions of points M in the plane P for which the ratio MB takes a minimum or maximum.48. (VIE 2) Let a be a number different from zero. For all integers n define Sn = an + a−n . Prove that if for some integer k both Sk and Sk+1 are integers, then for each integer n the number Sn is an integer.49. (VIE 3) Determine an equation of third degree with integral coefficients having roots sin 14 , sin 5π and sin −3π . π 14 1450. (YUG 1) Let m and n be natural numbers with m > n. Prove that 2(m − n)2 (m2 − n2 + 1) ≥ 2m2 − 2mn + 1.51. (YUG 2) There are n points on a flat piece of paper, any two of them at a distance of at least 2 from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals 3/2. Prove that there exist two vectors of equal length less than 1 and with their sum having a given direction, such that after a translation by either of these two vectors no points of the given set remain in the damaged area.52. (YUG 3) A fox stands in the center of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to u and v, respectively. Prove that: (a) If 2u > v, the fox can catch the rabbit, no matter how the rabbit moves. (b) If 2u ≤ v, the rabbit can always run away from the fox.3.16.3 Shortlisted Problems 1. I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam- inations
102 3 Problems11. II 5 (BUL 1)IMO4 Consider a partition of an 8 × 8 chessboard into p rectangles whose interiors are disjoint such that each of them has12. II 6 (USS 1) In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any nonallowed word.
3.17 IMO 1975 1033.17 The Seventeenth IMOBurgas–Sofia, Bulgaria, 19753.17.1 Contest Problems First Day (July 7)1. Let x1 ≥ x2 ≥ · · · ≥ xn and y1 ≥ y2 ≥ · · · ≥ yn be two n-tuples of numbers. Prove that n n (xi − yi )2 ≤ (xi − zi )2 i=1 i=1 is true when z1 , z2 , . . . , zn denote y1 , y2 , . . . , yn taken in another order.2. Let a1 , a2 , a3 , . . . be any infinite increasing sequence of positive integers. (For every integer i > 0, ai+1 > ai .) Prove that there are infinitely many m for which positive integers x, y, h, k can be found such that 0 < h < k < m and am = xah + yak .3. On the sides of an arbitrary triangle ABC, triangles BP C, CQA, and ARB are externally erected such that P BC = CAQ = 45◦ , BCP = QCA = 30◦ , ABR = BAR = 15◦ . Prove that QRP = 90◦ and QR = RP . Second Day (July 8)4. Let A be the sum of the digits of the number 44444444 and B the sum of the digits of the number A. Find the sum of the digits of the number B.5. Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)?6. The function f (x, y) is a homogeneous polynomial of the nth degree in x and y. If f (1, 0) = 1 and for all a, b, c, f (a + b, c) + f (b + c, a) + f (c + a, b) = 0, prove that f (x, y) = (x − 2y)(x + y)n−1 .3.17.2 Shortlisted Problems1. (FRA) There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions: (i) Every route includes exactly three ports;
106 3 Problems3.18 The Eighteenth IMOWienna–Linz, Austria, 19763.18.1 Contest Problems First Day (July 12)1. In a convex quadrangle with area 32 cm2 , the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to 16 cm. What is the length of the other diagonal?2. Let P1 (x) = x2 − 2, Pj (x) = P1 (Pj−1 (x)), j = 2, 3, . . . . Show that for arbitrary n, the roots of the equation Pn (x) = x are real and different from one another.3. A rectangular box can be filled completely with unit cubes. If one places cubes with volume 2 in the box such that their edges are parallel to the edges of the box, one can fill exactly 40% of the box. Determine all possible (interior) sizes of the box. Second Day (July 13)4. Find the largest number obtainable as the product of positive integers whose sum is 1976.5. Let a set of p equations be given, a11 x1 + · · · + a1q xq = 0, a21 x1 + · · · + a2q xq = 0, . . . ap1 x1 + · · · + apq xq = 0, with coefficients aij satisfying aij = −1, 0, or +1 for all i = 1, . . . , p and j = 1, . . . , q. Prove that if q = 2p, there exists a solution x1 , . . . , xq of this system such that all xj (j = 1, . . . , q) are integers satisfying |xj | ≤ q and xj = 0 for at least one value of j.6. For all positive integral n, un+1 = un (u2 −2)−u1 , u0 = 2, and u1 = 2 2 . n−1 1 Prove that 3 log2 [un ] = 2n − (−1)n , where [x] is the integral part of x.3.18.2 Longlisted Problems1. (BUL 1) (SL76-1).2. (BUL 2) Let P be a set of n points and S a set of l segments. It is known that:
3.18 IMO 1976 107 (i) No four points of P are coplanar. (ii) Any segment from S has its endpoints at P . (iii) There is a point, say g, in P that is the endpoint of a maximal number of segments from S and that is not a vertex of a tetrahedron having all its edges in S. 2 Prove that l ≤ n . 33. (BUL 3) (SL76-2).4. (BUL 4) Find all pairs of natural numbers (m, n) for which 2m · 3n + 1 is the square of some integer.5. (BUL 5) Let ABCDS be a pyramid with four faces and with ABCD as a base, and let a plane α through the vertex A meet its edges SB and SD at points M and N , respectively. Prove that if the intersection of the plane α with the pyramid ABCDS is a parallelogram, then SM · SN > BM · DN.6. (CZS 1) For each point X of a given polytope, denote by f (X) the sum of the distances of the point X from all the planes of the faces of the polytope. Prove that if f attains its maximum at an interior point of the polytope, then f is constant.7. (CZS 2) Let P be a fixed point and T a given triangle that contains the point P . Translate the triangle T by a given vector v and denote by T this new triangle. Let r, R, respectively, be the radii of the smallest disks centered at P that contain the triangles T , T , respectively. Prove that r + |v| ≤ 3R and find an example to show that equality can occur.8. (CZS 3) (SL76-3).9. (CZS 4) Find all (real) solutions of the system 3x1 − x2 − x3 − x5 = 0, −x1 + 3x2 − x4 − x6 = 0, −x1 + 3x3 − x4 − x7 = 0, −x2 − x3 + 3x4 − x8 = 0, −x1 + 3x5 − x6 − x7 = 0, −x2 − x5 + 3x6 − x8 = 0, −x3 − x5 + 3x7 − x8 = 0, −x4 − x6 − x7 + 3x8 = 0.
108 3 Problems10. (FIN 1) Show that the reciprocal of any number of the form 2(m2 + m + 1), where m is a positive integer, can be represented as a sum of consecutive terms in the sequence (aj )∞ , j=1 1 aj = . j(j + 1)(j + 2)11. (FIN 2) (SL76-9).12. (FIN 3) Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.13. (GBR 1a) (SL76-4).14. (GBR 1b) A sequence {un } of integers is defined by u1 = 2, u2 = u3 = 7, un+1 = un un−1 − un−2 , for n ≥ 3. Prove that for each n ≥ 1, un differs by 2 from an integral square.15. (GBR 2) Let ABC and A B C be any two coplanar triangles. Let L be a point such that AL BC, A L B C , and M, N similarly defined. The line BC meets B C at P , and similarly defined are Q and R. Prove that P L, QM , RN are concurrent.16. (GBR 3) Prove that there is a positive integer n such that the decimal representation of 7n contains a block of at least m consecutive zeros, where m is any given positive integer.17. (GBR 4) Show that there exists a convex polyhedron with all its vertices on the surface of a sphere√ √ all its faces congruent isosceles triangles and with whose ratio of sides are 3 : 3 : 2.18. (GDR 1) Prove that the number 191976 + 761976 : 4 (a) is divisible by the (Fermat) prime number F4 = 22 + 1; (b) is divisible by at least four distinct primes other than F4 .19. (GDR 2) For a positive integer n, let 6(n) be the natural number whose decimal representation consists of n digits 6. Let us define, for all natural numbers m, k with 1 ≤ k ≤ m, m 6(m) · 6(m−1) · · · 6(m−k+1) = . k 6(1) · 6(2) · · · 6(k) m Prove that for all m, k, is a natural number whose decimal repre- k sentation consists of exactly k(m + k − 1) − 1 digits.20. (GDR 3) Let (an ), n = 0, 1, . . ., be a sequence of real numbers such that a0 = 0 and
110 3 Problems29. (POL 1b) (SL76-7).30. (POL 2) Prove that if P (x) = (x−a)k Q(x), where k is a positive integer, a is a nonzero real number, Q(x) is a nonzero polynomial, then P (x) has at least k + 1 nonzero coefficients.31. (POL 3) Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.32. (POL 4) We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.33. (SWE 1) A finite set of points P in the plane has the following prop- erty: Every line through two points in P contains at least one more point belonging to P . Prove that all points in P lie on a straight line.34. (SWE 2) Let {an }∞ and {bn }∞ be two sequences determined by the 0 0 recursion formulas an+1 = an + bn , bn+1 = 3an + bn , n = 0, 1, 2, . . . , and the initial values a0 = b0 = 1. Prove that there exists a uniquely determined constant c such that n|can −bn | < 2 for all nonnegative integers n.35. (SWE 3) (SL76-8).36. (USA 1) Three concentric circles with common center O are cut by a common chord in successive points A, B, C. Tangents drawn to the circles at the points A, B, C enclose a triangular region. If the distance from point O to the common chord is equal to p, prove that the area of the region enclosed by the tangents is equal to AB · BC · CA . 2p37. (USA 2) From a square board 11 squares long and 11 squares wide, the central square is removed. Prove that the remaining 120 squares cannot be covered by 15 strips each 8 units long and one unit wide. √ √38. (USA 3) Let x = a + b, where a and b are natural numbers, x is not an integer, and x < 1976. Prove that the fractional part of x exceeds 10−19.76 .39. (USA 4) In ABC, the inscribed circle is tangent to side BC at X. Segment AX is drawn. Prove that the line joining the midpoint of segment
3.18 IMO 1976 111 AX to the midpoint of side BC passes through the center I of the inscribed circle.40. (USA 5) Let g(x) be a fixed polynomial and define f (x) by f (x) = x2 + xg(x3 ). Show that f (x) is not divisible by x2 − x + 1.41. (USA 6) (SL76-10).42. (USS 1) For a point O inside a triangle ABC, denote by A1 , B1 , C1 the respective intersection points of AO, BO, CO with the corresponding AO BO CO sides. Let n1 = A1 O , n2 = B1 O , n3 = C1 O . What possible values of n1 , n2 , n3 can all be positive integers?43. (USS 2) Prove that if for a polynomial P (x, y) we have P (x − 1, y − 2x + 1) = P (x, y), then there exists a polynomial Φ(x) with P (x, y) = Φ(y − x2 ). √44. (USS 3) A circle of radius 1 rolls around a circle of radius 2. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made n circuits around the bigger one?45. (USS 4) We are given n (n ≥ 5) circles in a plane. Suppose that every three of them have a common point. Prove that all n circles have a common point.46. (USS 5) For a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, prove the inequality a4 + b4 + c4 + d4 + 2abcd ≥ a2 b2 + a2 c2 + a2 d2 + b2 c2 + b2 d2 + c2 d2 .47. (VIE 1) (SL76-11).48. (VIE 2) (SL76-12).49. (VIE 3) Determine whether there exist 1976 nonsimilar triangles with angles α, β, γ, each of them satisfying the relations sin α + sin β + sin γ 12 12 = and sin α sin β sin γ = . cos α + cos β + cos γ 7 2550. (VIE 4) Find a function f (x) defined for all real values of x such that for all x, f (x + 2) − f (x) = x2 + 2x + 4, and if x ∈ [0, 2), then f (x) = x2 .51. (YUG 1) Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.
3.18 IMO 1976 113 7. (POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given number a ∈ (0, 1) we define a map T : I → I by the formula x + (1 − a) if 0 < x ≤ a, T (x, y) = x−a if a < x ≤ 1. Show that for every interval J ⊂ I there exists an integer n > 0 such that T n (J) ∩ J = ∅. 8. (SWE 3) Let P be a polynomial with real coefficients such that P (x) > 0 if x > 0. Prove that there exist polynomials Q and R with nonnegative coefficients such that P (x) = Q(x) if x > 0. R(x) 9. (FIN 2)IMO2 Let P1 (x) = x2 − 2, Pj (x) = P1 (Pj−1 (x)), j = 2, 3, . . . . Show that for arbitrary n the roots of the equation Pn (x) = x are real and different from one another.10. (USA 6)IMO4 Find the largest number obtainable as the product of pos- itive integers whose sum is 1976.11. (VIE 1) Prove that there exist infinitely many positive integers n such that the decimal representation of 5n contains a block of 1976 consecutive zeros.12. (VIE 2) The polynomial 1976(x+x2 +· · ·+xn ) is decomposed into a sum of polynomials of the form a1 x + a2 x2 + . . . + an xn , where a1 , a2 , · · · , an are distinct positive integers not greater than n. Find all values of n for which such a decomposition is possible.
114 3 Problems3.19 The Nineteenth IMOBelgrade–Arandjelovac, Yugoslavia, July 1–13, 19773.19.1 Contest Problems First Day (July 6)1. Equilateral triangles ABK, BCL, CDM , DAN are constructed inside the square ABCD. Prove that the midpoints of the four segments KL, LM , M N , N K and the midpoints of the eight segments AK, BK, BL, CL, CM , DM , DN , AN are the twelve vertices of a regular dodecagon.2. In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Deter- mine the maximum number of terms in the sequence.3. Let n be a given integer greater than 2, and let Vn be the set of integers 1 + kn, where k = 1, 2, . . . . A number m ∈ Vn is called indecomposable in Vn if there do not exist numbers p, q ∈ Vn such that pq = m. Prove that there exists a number r ∈ Vn that can be expressed as the product of elements indecomposable in Vn in more than one way. (Expressions that differ only in order of the elements of Vn will be considered the same.) Second Day (July 7)4. Let a, b, A, B be given constant real numbers and f (x) = 1 − a cos x − b sin x − A cos 2x − B sin 2x. Prove that if f (x) ≥ 0 for all real x, then a2 + b 2 ≤ 2 and A2 + B 2 ≤ 1.5. Let a and b be natural numbers and let q and r be the quotient and remainder respectively when a2 + b2 is divided by a + b. Determine the numbers a and b if q 2 + r = 1977.6. Let f : N → N be a function that satisfies the inequality f (n+1) > f (f (n)) for all n ∈ N. Prove that f (n) = n for all natural numbers n.3.19.2 Longlisted Problems1. (BUL 1) A pentagon ABCDE inscribed in a circle for which BC < CD and AB < DE is the base of a pyramid with vertex S. If AS is the longest edge starting from S, prove that BS > CS.2. (BUL 2) (SL77-1).
3.19 IMO 1977 115 3. (BUL 3) In a company of n persons, each person has no more than d acquaintances, and in that company there exists a group of k persons, k ≥ d, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than [n2 /4]. 4. (BUL 4) We are given n points in space. Some pairs of these points are connected by line segments so that the number of segments equals [n2 /4], and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle. 5. (CZS 1) (SL77-2). 6. (CZS 2) Let x1 , x2 , . . . , xn (n ≥ 1) be real numbers such that 0 ≤ xj ≤ π, n j = 1, 2, . . . , n. Prove that if j=1 (cos xj + 1) is an odd integer, then n j=1 sin xj ≥ 1. 7. (CZS 3) Prove the following assertion: If c1 , c2 , . . . , cn (n ≥ 2) are real numbers such that (n − 1)(c2 + c2 + · · · + c2 ) = (c1 + c2 + · · · + cn )2 , 1 2 n then either all these numbers are nonnegative or all these numbers are nonpositive. 8. (CZS 4) A hexahedron ABCDE is made of two regular congruent tetra- hedra ABCD and ABCE. Prove that there exists only one isometry Z that maps points A, B, C, D, E onto B, C, A, E, D, respectively. Find all points X on the surface of hexahedron whose distance from Z(X) is minimal. 9. (CZS 5) Let ABCD be a regular tetrahedron and Z an isometry map- ping A, B, C, D into B, C, D, A, respectively. Find the set M of all points X of the face ABC whose distance from Z(X) is equal to a given number t. Find necessary and sufficient conditions for the set M to be nonempty.10. (FRG 1) (SL77-3).11. (FRG 2) Let n and z be integers greater than 1 and (n, z) = 1. Prove: (a) At least one of the numbers zi = 1+z +z 2 +· · ·+z i , i = 0, 1, . . . , n−1, is divisible by n. (b) If (z −1, n) = 1, then at least one of the numbers zi , i = 0, 1, . . . , n−2, is divisible by n.12. (FRG 3) Let z be an integer > 1 and let M be the set of all numbers of the form zk = 1 + z + · · · + z k , k = 0, 1, . . . . Determine the set T of divisors of at least one of the numbers zk from M .13. (FRG 4) (SL77-4).14. (FRG 5) (SL77-5).
116 3 Problems15. (GDR 1) Let n be an integer greater than 1. In the Cartesian coordinate system we consider all squares with integer vertices (x, y) such that 1 ≤ x, y ≤ n. Denote by pk (k = 0, 1, 2, . . . ) the number of pairs of points that are vertices of exactly k such squares. Prove that k (k − 1)pk = 0.16. (GDR 2) (SL77-6).17. (GDR 3) A ball K of radius r is touched from the outside by mutually equal balls of radius R. Two of these balls are tangent to each other. Moreover, for two balls K1 and K2 tangent to K and tangent to each other there exist two other balls tangent to K1 , K2 and also to K. How many balls are tangent to K? For a given r determine R.18. (GDR 4) Given an isosceles triangle ABC with a right angle at C, construct the center M and radius r of a circle cutting on segments AB, BC, CA the segments DE, F G, and HK, respectively, such that ∠DM E + ∠F M G + ∠HM K = 180◦ and DE : F G : HK = AB : BC : CA.19. (GBR 1) Given any integer m > 1 prove that there exist infinitely many positive integers n such that the last m digits of 5n are a sequence am , am−1 , . . . , a1 = 5 (0 ≤ aj < 10) in which each digit except the last is of opposite parity to its successor (i.e., if ai is even, then ai−1 is odd, and if ai is odd, then ai−1 is even).20. (GBR 2) (SL77-7).21. (GBR 3) Given that x1 +x2 +x3 = y1 +y2 +y3 = x1 y1 +x2 y2 +x3 y3 = 0, prove that x21 2 y1 2 + 2 2 = 3. x2 + x2 + x2 1 2 3 2 y1 + y2 + y322. (GBR 4) (SL77-8).23. (HUN 1) (SL77-9).24. (HUN 2) Determine all real functions f (x) that are defined and contin- uous on the interval (−1, 1) and that satisfy the functional equation f (x) + f (y) f (x + y) = (x, y, x + y ∈ (−1, 1)). 1 − f (x)f (y)25. (HUN 3) Prove the identity n n n n (z + a) = z + a (a − kb)k−1 (z + kb)n−k . k k=126. (NET 1) Let p be a prime number greater than 5. Let V be the collection of all positive integers n that can be written in the form n = kp + 1 or n = kp − 1 (k = 1, 2, . . . ). A number n ∈ V is called indecomposable in V if it is impossible to find k, l ∈ V such that n = kl. Prove that there exists
118 3 Problems ⎡ ⎤2 ⎛ ⎞⎡ ⎤ n n n ⎣ mj (aj + bj + cj )⎦ > 3 ⎝ mj ⎠ ⎣ mj (aj bj + bj cj + cj aj )⎦ . j=1 j=1 j=139. (ROM 5) Consider 37 distinct points in space, all with integer coordi- nates. Prove that we may find among them three distinct points such that their barycenter has integers coordinates.40. (SWE 1) The numbers 1, 2, 3, . . . , 64 are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size 2 × 2. Prove that there are at least three such squares for which the sum of the 4 numbers contained exceeds 100.41. (SWE 2) A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers 1, 2, 3, . . . , N are marked, and on the ring N integers a1 , a2 , . . . , aN of sum 1 are marked (see the figure). The · · · a4 ring can be turned into N differ- · · · 4 a3 3 a2 ent positions in which the numbers 2 on the disk and on the ring match 1 a1 N aN each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of N products. In this way a sum is obtained for every position of the ring. Prove that the N sums are different.42. (SWE 3) The sequence an,k , k = 1, 2, 3, . . . , 2n , n = 0, 1, 2, . . . , is defined by the following recurrence formula: 1 3 a1 = 2, an,k = 2a3 n−1,k , an,k+2n−1 = a 2 n−1,k for k = 1, 2, 3, . . . , 2n−1 , n = 0, 1, 2, . . . . Prove that the numbers an,k are all different.43. (FIN 1) Evaluate n S= k(k + 1) · · · (k + p), k=1 where n and p are positive integers.44. (FIN 2) Let E be a finite set of points in space such that E is not contained in a plane and no three points of E are collinear. Show that E contains the vertices of a tetrahedron T = ABCD such that T ∩ E = {A, B, C, D} (including interior points of T ) and such that the projection of A onto the plane BCD is inside a triangle that is similar to the triangle BCD and whose sides have midpoints B, C, D.
3.19 IMO 1977 11945. (FIN 2 ) (SL77-14).46. (FIN 3) Let f be a strictly increasing function defined on the set of real numbers. For x real and t positive, set f (x + t) − f (x) g(x, t) = . f (x) − f (x − t) Assume that the inequalities 2−1 < g(x, t) < 2 hold for all positive t if x = 0, and for all t ≤ |x| otherwise. Show that 14−1 < g(x, t) < 14 for all real x and positive t.47. (USS 1) A square ABCD is given. A line passing through A intersects CD at Q. Draw a line parallel to AQ that intersects the boundary of the square at points M and N such that the area of the quadrilateral AM N Q is maximal.48. (USS 2) The intersection of a plane with a regular tetrahedron with edge a is a quadrilateral with perimeter P . Prove that 2a ≤ P ≤ 3a.49. (USS 3) Find all pairs of integers (p, q) for which all roots of the trino- mials x2 + px + q and x2 + qx + p are integers.50. (USS 4) Determine all positive integers n for which there exists a poly- nomial Pn (x) of degree n with integer coefficients that is equal to n at n different integer points and that equals zero at zero.51. (USS 5) Several segments, which we shall call white, are given, and the sum of their lengths is 1. Several other segments, which we shall call black, are given, and the sum of their lengths is 1. Prove that every such system of segments can be distributed on the segment that is 1.51 long in the following way: Segments of the same color are disjoint, and segments of different colors are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is 1.49 long.52. (USA 1) Two perpendicular chords are drawn through a given interior point P of a circle with radius R. Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from P to the center of the circle is kR.53. (USA 2) Find all pairs of integers a and b for which 7a + 14b = 5a2 + 5ab + 5b2 .
120 3 Problems54. (USA 3) If 0 ≤ a ≤ b ≤ c ≤ d, prove that ab bc cd da ≥ ba cb dc ad .55. (USA 4) Through a point O on the diagonal BD of a parallelogram ABCD, segments M N parallel to AB, and P Q parallel to AD, are drawn, with M on AD, and Q on AB. Prove that diagonals AO, BP , DN (ex- tended if necessary) will be concurrent.56. (USA 5) The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.57. (VIE 1) (SL77-15).58. (VIE 2) Prove that for every triangle the following inequality holds: ab + bc + ca π ≥ cot , 4S 6 where a, b, c are lengths of the sides and S is the area of the triangle.59. (VIE 3) (SL77-16).60. (VIE 4) Suppose x0 , x1 , . . . , xn are integers and x0 > x1 > · · · > xn . Prove that at least one of the numbers |F (x0 )|, |F (x1 )|, |F (x2 )|, . . . , |F (xn )|, where F (x) = xn + a1 xn−1 + · · · + an , ai ∈ R, i = 1, . . . , n, n! is greater than 2n .3.19.3 Shortlisted Problems 1. (BUL 2)IMO6 Let f : N → N be a function that satisfies the inequality f (n + 1) > f (f (n)) for all n ∈ N. Prove that f (n) = n for all natural numbers n. 2. (CZS 1) A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let k be a circle with radius r ≥ 2, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle k that has a neighboring point lying outside k. Similarly, an exterior boundary point is a lattice point lying outside the circle k that has a neighboring point lying inside k. Prove that there are four more exterior boundary points than interior boundary points. 3. (FRG 1)IMO5 Let a and b be natural numbers and let q and r be the quotient and remainder respectively when a2 + b2 is divided by a + b. Determine the numbers a and b if q 2 + r = 1977.
122 3 Problems11. (NET 3) Let n be an integer greater than 1. Define xi + yi n x1 = n, y1 = 1, xi+1 = , yi+1 = for i = 1, 2, . . . , 2 xi+1 where [z] denotes the largest integer less than or equal to z. Prove that √ min{x1 , x2 , . . . xn } = [ n].12. (NET 4)IMO1 On the sides of a square ABCD one constructs inwardly equilateral triangles ABK, BCL, CDM , DAN . Prove that the midpoints of the four segments KL, LM , M N , N K, together with the midpoints of the eight segments AK, BK, BL, CL, CM , DM , DN , AN , are the 12 vertices of a regular dodecagon.13. (POL 4) Let B be a set of k sequences each having n terms equal to 1 or −1. The product of two such sequences (a1 , a2 , . . . , an ) and (b1 , b2 , . . . , bn ) is defined as (a1 b1 , a2 b2 , . . . , an bn ). Prove that there exists a sequence (c1 , c2 , . . . , cn ) such that the intersection of B and the set containing all sequences from B multiplied by (c1 , c2 , . . . , cn ) contains at most k 2 /2n sequences.14. (FIN 2') Let E be a finite set of points such that E is not contained in a plane and no three points of E are collinear. Show that at least one of the following alternatives holds: (i) E contains five points that are vertices of a convex pyramid having no other points in common with E; (ii) some plane contains exactly three points from E.15. (VIE 1)IMO2 The length of a finite sequence is defined as the number of terms of this sequence. Determine the maximal possible length of a finite sequence that satisfies the following condition: The sum of each seven successive terms is negative, and the sum of each eleven successive terms is positive.16. (VIE 3) Let E be a set of n points in the plane (n ≥ 3) whose co- ordinates are integers such that any three points from E are vertices of a nondegenerate triangle whose centroid doesn't have both coordinates integers. Determine the maximal n.
3.20 IMO 1978 1233.20 The Twentieth IMOBucharest, Romania, 19783.20.1 Contest Problems First Day (July 6)1.2. Given any point P in the interior of a sphere with radius R, three mutu- ally perpendicular segments P A, P B, P C are drawn terminating on the sphere and having one common vertex in P . Consider the rectangular par- allelepiped of which P A, P B, P C are coterminal edges. Find the locus of the point Q that is diagonally opposite P in the parallelepiped when P and the sphere are fixed.3. Let {f (n)} be a strictly increasing sequence of positive integers: 0 < f (1) < f (2) < f (3) < . . . . Of the positive integers not belonging to the sequence, the nth in order of magnitude is f (f (n)) + 1. Determine f (240). Second day (July 7)4.5. Let ϕ : {1, 2, 3, . . . } → {1, 2, 3, . . . } be injective. Prove that for all n, n n ϕ(k) 1 ≥ . k2 k k=1 k=16.3.20.2 Longlisted Problems1. (BUL 1) (SL78-1).2. (BUL 2) If f (x) = (x + 2x2 + · · · + nxn )2 = a2 x2 + a3 x3 + · · · + a2n x2n , prove that
3.20 IMO 1978 125 (ii) p(x, y) is an increasing and q(x) is a decreasing function of y for every fixed x. (iii) p(x, 0) = q(x, 0) for every x and p(0, 0) = 0. Show that the simultaneous equations p(x, y) = a, q(x, y) = b have a unique solution in the set x ≥ 0, y ≥ 0 for all a, b satisfying 0 ≤ b ≤ a but lack a solution in the same set if a < b.15. (FRA 1) Prove that for every positive integer n coprime to 10 there exists a multiple of n that does not contain the digit 1 in its decimal representation.16. (FRA 2) (SL78-6).17. (FRA 3) (SL78-17).18. (FRA 4) Given a natural number n, prove that the number M (n) of √ points with integer coordinates inside the circle (O(0, 0), n) satisfies √ √ πn − 5 n + 1 < M (n) < πn + 4 n + 1.19. (FRA 5) (SL78-7).20. (GBR 1) Let O be the center of a circle. Let OU, OV be perpendicular radii of the circle. The chord P Q passes through the midpoint M of U V . Let W be a point such that P M = P W , where U, V, M, W are collinear. Let R be a point such that P R = M Q, where R lies on the line P W . Prove that M R = U V . Alternative version: A circle S is given with center O and radius r. Let r M be a point whose distance from O is √2 . Let P M Q be a chord of S. −→ − −→ − The point N is defined by P N = M Q. Let R be the reflection of N by √ the line through P that is parallel to OM . Prove that M R = 2r.21. (GBR 2) A circle touches the sides AB, BC, CD, DA of a square at points K, L, M, N respectively, and BU, KV are parallel lines such that U is on DM and V on DN . Prove that U V touches the circle.22. (GBR 3) Two nonzero integers x, y (not necessarily positive) are such 2 +y 2 that x + y is a divisor of x2 + y 2 , and the quotient xx+y is a divisor of 1978. Prove that x = y.23. (GBR 4) (SL78-8).24. (GBR 5) (SL78-9).25. (GDR 1) Consider a polynomial P (x) = ax2 + bx + c with a > 0 that has two real roots x1 , x2 . Prove that the absolute values of both roots are less than or equal to 1 if and only if a + b + c ≥ 0, a − b + c ≥ 0, and a − c ≥ 0.26. (GDR 2) (SL78-5).
128 3 Problems (a) Pn (x) satisfies the identity Pn (x) − xPn−1 (x) + 1 Pn−2 (x) ≡ 0. 4 (b) Pn (x) is a polynomial in x of degree n.48. (VIE 2) (SL78-14).49. (VIE 3) Let A, B, C, D be four arbitrary distinct points in space. (a) Prove that using the segments AB + CD, AC + BD and AD + BC it is always possible to construct a triangle T that is nondegenerate and has no obtuse angle. (b) What should these four points satisfy in order for the triangle T to be right-angled?50. (VIE 4) A variable tetrahedron ABCD has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal (BC = DA, CA = DB, AB = DC); and the vertices A, B, C lie respectively on three fixed spheres with the same center P and radii 3, 4, 12. What is the maximal length of P D?51. (VIE 5) Find the relations among the angles of the triangle ABC whose altitude AH and median AM satisfy ∠BAH = ∠CAM .52. (YUG 1) (SL78-15).53. (YUG 2) (SL78-16).54. (YUG 3) Let p, q and r be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes α, β, and γ, containing p, q, and r respectively, that are perpendicular to each other (α ⊥ β, β ⊥ γ, γ ⊥ α).3.20.3 Shortlisted Problems 1. (BUL 1) The set M = {1, 2, . . . , 2n} is partitioned into k nonintersecting subsets M1 , M2 , . . . , Mk , where n ≥ k 3 + k. Prove that there exist even numbers 2j1 , 2j2 , . . . , 2jk+1 in M that are in one and the same subset Mi (1 ≤ i ≤ k) such that the numbers 2j1 − 1, 2j2 − 1, . . . , 2jk+1 − 1 are also in one and the same subset Mj (1 ≤ j ≤ k). 2. (BUL 4) Two identically oriented equilateral triangles, ABC with center S and A B C, are given in the plane. We also have A = S and B = S. If M is the midpoint of A B and N the midpoint of AB , prove that the triangles SB M and SA N are similar. 3. (CUB 3)IMO1 4. (CZS 2) Let T1 be a triangle having a, b, c as lengths of its sides and let T2 be another triangle having u, v, w as lengths of its sides. If P, Q are the areas of the two triangles, prove that
3.20 IMO 1978 129 16P Q ≤ a2 (−u2 + v 2 + w2 ) + b2 (u2 − v 2 + w2 ) + c2 (u2 + v 2 − w2 ). When does equality hold? 5. (GDR 2) For every integer d ≥ 1, let Md be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference d, having at least two terms and consisting of positive integers. Let A = M1 , B = M2 {2}, C = M3 . Prove that every c ∈ C may be written in a unique way as c = ab with a ∈ A, b ∈ B. 6. (FRA 2)IMO5 Let ϕ : {1, 2, 3, . . . } → {1, 2, 3, . . . } be injective. Prove that for all n, n n ϕ(k) 1 2 ≥ . k k k=1 k=1 7. (FRA 5) We consider three distinct half-lines Ox, Oy, Oz in a plane. Prove the existence and uniqueness of three points A ∈ Ox, B ∈ Oy, C ∈ Oz such that the perimeters of the triangles OAB, OBC, OCA are all equal to a given number 2p > 0. 8. (GBR 4) Let S be the set of all the odd positive integers that are not multiples of 5 and that are less than 30m, m being an arbitrary positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two different integers, one of which divides the other? 9. (GBR 5)IMO3 Let {f (n)} be a strictly increasing sequence of positive integers: 0 < f (1) < f (2) < f (3) < · · · . Of the positive integers not belonging to the sequence, the nth in order of magnitude is f (f (n)) + 1. Determine f (240).10. (NET 1)IMO611. (SWE 2) A function f : I → R, defined on an interval I, is called concave if f (θx + (1 − θ)y) ≥ θf (x) + (1 − θ)f (y) for all x, y ∈ I and 0 ≤ θ ≤ 1. Assume that the functions f1 , . . . , fn , having all nonnegative values, are concave. Prove that the function (f1 f2 . . . fn )1/n is concave.12. (USA 1)IMO413. (USA 6)IMO2 Given any point P in the interior of a sphere with ra- dius R, three mutually perpendicular segments P A, P B, P C are drawn terminating on the sphere and having one common vertex in P . Con- sider the rectangular parallelepiped of which P A, P B, P C are coterminal
130 3 Problems edges. Find the locus of the point Q that is diagonally opposite P in the parallelepiped when P and the sphere are fixed.14. (VIE 2) Prove that it is possible to place 2n(2n + 1) parallelepipedic (rectangular) pieces of soap of dimensions 1 × 2 × (n + 1) in a cubic box with edge 2n + 1 if and only if n is even or n = 1. Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.15. (YUG 1) Let p be a prime and A = {a1 , . . . , ap−1 } an arbitrary subset of the set of natural numbers such that none of its elements is divisible by p. Let us define a mapping f from P(A) (the set of all subsets of A) to the set P = {0, 1, . . . , p − 1} in the following way: k (i) if B = {ai1 , . . . , aik } ⊂ A and j=1 aij ≡ n (mod p), then f (B) = n, (ii) f (∅) = 0, ∅ being the empty set. Prove that for each n ∈ P there exists B ⊂ A such that f (B) = n.16. (YUG 2) Determine all the triples (a, b, c) of positive real numbers such that the system ax + by − cz = 0, a 1 − x2 + b 1 − y2 − c 1 − z 2 = 0, is compatible in the set of real numbers, and then find all its real solutions.17. (FRA 3) Prove that for any positive integers x, y, z with xy − z 2 = 1 one can find nonnegative integers a, b, c, d such that x = a2 + b2 , y = c2 + d2 , z = ac + bd. Set z = (2q)! to deduce that for any prime number p = 4q + 1, p can be represented as the sum of squares of two integers.
3.21 IMO 1979 1313.21 The Twenty-First IMOLondon, United Kingdom, 19793.21.1 Contest Problems First Day (July 2)1. Given that 1 1 1 1 1 p 1− + − + ···− + = , 2 3 4 1318 1319 q where p and q are natural numbers having no common factor, prove that p is divisible by 1979.2. A pentagonal prism A1 A2 . . . A5 B1 B2 . . . B5 is given. The edges, the diag- onals of the lateral walls,3. Second Day (July 3)4. Given a point P in a given plane π and also a given point Q not in π, determine all points R in π such that QP +P R is a maximum. QR5. The nonnegative real numbers x1 , x2 , x3 , x4 , x5 , a satisfy the following re- lations: 5 5 5 ixi = a, i3 xi = a2 , i5 xi = a3 . i=1 i=1 i=1 What are the possible values of a?6. Let S and F be two opposite vertices of a regular octagon. A counter starts at S and each second is moved to one of the two neighboring 2
3.21 IMO 1979 133 Variant. Prove that a pyramid A1 . . . A2k+1 S with equal space angles between adjacent lateral walls is regular if there exists a sphere tangent to all its edges.12. (BUL 3) (SL79-4).13. (BUL 4) The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let M be an arbitrary set of n squares of this net. Prove that it is possible to choose no fewer than n/4 squares of M in such a way that no two of them have a common point.14. (CZS 1) Let S be a set of n2 + 1 closed intervals (n a positive integer). Prove that at least one of the following assertions holds: (i) There exists a subset S of n + 1 intervals from S such that the inter- section of the intervals in S is nonempty. (ii) There exists a subset S of n + 1 intervals from S such that any two of the intervals in S are disjoint.15. (CZS 2) (SL79-5).16. (CZS 3) Let Q be a square with side length 6. Find the smallest integer n such that in Q there exists a set S of n points with the property that any square with side 1 completely contained in Q contains in its interior at least one point from S.17. (CZS 4) (SL79-6).18. (FIN 1) Show that for no integers a ≥ 1, n ≥ 1 is the sum 1 1 1 1+ + + ··· + 1 + a 1 + 2a 1 + na an integer.19. (FIN 2) For k = 1, 2, . . . consider the k-tuples (a1 , a2 , . . . , ak ) of positive integers such that a1 + 2a2 + · · · + kak = 1979. Show that there are as many such k-tuples with odd k as there are with even k.20. (FIN 3) (SL79-10).21. (FRA 1) Let E be the set of all bijective mappings from R to R satisfying (∀t ∈ R) f (t) + f −1 (t) = 2t, where f −1 is the mapping inverse to f . Find all elements of E that are monotonic mappings.22. (FRA 2) Consider two quadrilaterals ABCD and A B C D in an affine Euclidian plane such that AB = A B , BC = B C , CD = C D , and DA = D A . Prove that the following two statements are true:
134 3 Problems (a) If the diagonals BD and AC are mutually perpendicular, then the diagonals B D and A C are also mutually perpendicular. (b) If the perpendicular bisector of BD intersects AC at M , and that MA M A of B D intersects A C at M , then MC = M C (if M C = 0 then M C = 0).23. (FRA 3) Consider the set E consisting of pairs of integers (a, b), with a ≥ 1 and b ≥ 1, that satisfy in the decimal system the following properties: (i) b is written with three digits, as α2 α1 α0 , α2 = 0; (ii) a is written as βp . . . β1 β0 for some p; (iii) (a + b)2 is written as βp . . . β1 β0 α2 α1 α0 . Find the elements of E.24. (FRA 4) Let a and b be coprime integers, greater than or equal to 1. Prove that all integers n greater than or equal to (a − 1)(b − 1) can be written in the form: n = ua + vb, with (u, v) ∈ N × N.25. (FRG 1) (SL79-7).26. (FRG 2) Let n be a natural number. If 4n + 2n + 1 is a prime, prove that n is a power of three.27. (FRG 3) (SL79-8).28. (FRG 4) (SL79-9).29. (GDR 1) (SL79-11).30. (GDR 2) Let M be a set of points in a plane with at least two elements. Prove that if M has two axes of symmetry g1 and g2 intersecting at an angle α = qπ, where q is irrational, then M must be infinite.31. (GDR 3) (SL79-12).32. (GDR 4) Let n, k ≥ 1 be natural numbers. Find the number A(n, k) of solutions in integers of the equation |x1 | + |x2 | + · · · + |xk | = n.33. (GRE 1) (SL79-13).34. (GRE 2) Notice that in the fraction 16 we can perform a simplification 64 as 16 = 1 obtaining a correct equality. Find all fractions whose numer- 64 4 ators and denominators are two-digit positive integers for which such a simplification is correct.35. (GRE 3) Given a sequence (an ), with a1 = 4 and an+1 = a2 −2 (∀n ∈ N), n prove that there is a triangle with side lengths an − 1, an , an + 1, and that its area is equal to an integer.
3.21 IMO 1979 13536. (GRE 4) A regular tetrahedron A1 B1 C1 D1 is inscribed in a regular tetrahedron ABCD, where A1 lies in the plane BCD, B1 in the plane ACD, etc. Prove that A1 B1 ≥ AB/3.37. (GRE 5) (SL79-14).38. (HUN 1) Prove the following statement: If a polynomial f (x) with real coefficients takes only nonnegative values, then there exists a positive integer n and polynomials g1 (x), g2 (x), . . . , gn (x) such that f (x) = g1 (x)2 + g2 (x)2 + · · · + gn (x)2 .39. (HUN 2) A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of n days of walking from the camp, under the following conditions: (i) Each member of the expedition can pick up at most 3 liters of water. (ii) Each member must drink one liter of water every day spent in the desert. (iii) All the members must return to the camp. How much water do they need (at least) in order to do that?40. (HUN 3) A polynomial P (x) has degree at most 2k, where k = 0, 1, 2, . . . . Given that for an integer i, the inequality −k ≤ i ≤ k implies |P (i)| ≤ 1, prove that for all real numbers x, with −k ≤ x ≤ k, the following inequality holds: 2k |P (x)| < (2k + 1) . k41. (HUN 4) Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.42. (HUN 5) Let a quadratic polynomial g(x) = ax2 + bx + c be given and an integer n ≥ 1. Prove that there exists at most one polynomial f (x) of nth degree such that f (g(x)) = g(f (x)).43. (ISR 1) Let a, b, c denote the lengths of the sides BC, CA, AB, respec- tively, of a triangle ABC. If P is any point on the circumference of the circle inscribed in the triangle, show that aP A2 +bP B 2 +cP C 2 is constant.44. (ISR 2) (SL79-15).45. (ISR 3) For any positive integer n we denote by F (n) the number of ways in which n can be expressed as the sum of three different positive integers, without regard to order. Thus, since 10 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2, we have F (10) = 4. Show that F (n) is even if n ≡ 2 or 4 (mod 6), but odd if n is divisible by 6.46. (ISR 4) (SL79-16).
3.21 IMO 1979 137 √ √56. (ROM 3) Show that for every natural number n, n 2 − [n 2] > 1 √ √ 2 2n and that for every ε > 0 there exists a natural number n with n 2 − √ [n 2] < 2n1 2 + ε. √57. (ROM 4) Let M be a set, and A, B, C given subsets of M . Find a necessary and sufficient condition for the existence of a set X ⊂ M for which (X ∪ A) (X ∩ B) = C. Describe all such sets X.58. (ROM 5) Prove that there exists a natural number k0 such that for every natural number k > k0 we may find a finite number of lines in the plane, not all parallel to one of them, that divide the plane exactly in k regions. Find k0 .59. (SWE 1) Determine the maximum value of x2 y 2 z 2 w when x, y, z, w ≥ 0 and 2x + xy + z + yzw = 1.60. (SWE 2) (SL79-20).61. (SWE 3) Let a1 ≤ a2 ≤ · · · ≤ an and b1 ≤ b2 ≤ · · · ≤ bn be two m m sequences such that k=1 ak ≥ k=1 bk for all m ≤ n with equality for m = n. Let f be a convex function defined on the real numbers. Prove that n n f (ak ) ≤ f (bk ). k=1 k=162. (SWE 4) T is a given triangle with vertices P1 , P2 , P3 . Consider an arbi- trary subdivision of T into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex V of the subtriangles there is assigned a number n(V ) according to the following rules: (i) If V = Pi , then n(V ) = i. (ii) If V lies on the side Pi Pj of T , then n(V ) = i or j. (iii) If V lies inside the triangle T , then n(V ) is any of the numbers 1,2,3. Prove that there exists at least one subtriangle whose vertices are num- bered 1, 2, and 3.63. (USA 1) If a1 , a2 , . . . , an denote the lengths of the sides of an arbitrary n-gon, prove that a1 a2 an n 2≥ + + ··· + ≥ , s − a1 s − a2 s − an n−1 where s = a1 + a2 + · · · + an .64. (USA 2) From point P on arc BC of the circumcircle about triangle ABC, P X is constructed perpendicular to BC, P Y is perpendicular to AC, and P Z perpendicular to AB (all extended if necessary). Prove that BC AC AB = + . PX PY PZ
138 3 Problems65. (USA 3) Given f (x) ≤ x for all real x and f (x + y) ≤ f (x) + f (y) for all real x, y, prove that f (x) = x for all x.66. (USA 4) (SL79-23).67. (USA 5) (SL79-24).68. (USA 6) (SL79-25).69. (USS 1) (SL79-21).70. (USS 2) There are 1979 equilateral triangles: T1 , T2 , . . . , T1979 . A side of triangle Tk is equal to 1/k, k = 1, 2, . . . , 1979. At what values of a number a can one place all these triangles into the equilateral triangle with side length a so that they don't intersect (points of contact are allowed)?71. (USS 3) (SL79-22).72. (VIE 1) Let f (x) be a polynomial with integer coefficients. Prove that if f (x) equals 1979 for four different integer values of x, then f (x) cannot be equal to 2 × 1979 for any integral value of x.73. (VIE 2) In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?74. (VIE 3) Given an equilateral triangle ABC of side a in a plane, let M be a point on the circumcircle of the triangle. Prove that the sum s = M A4 + M B 4 + M C 4 is independent of the position of the point M on the circle, and determine that constant value as a function of a.75. (VIE 4) Given an equilateral triangle ABC, let M be an arbitrary point in space. (a) Prove that one can construct a triangle from the segments M A, M B, M C. (b) Suppose that P and Q are two points symmetric with respect to the center O of ABC. Prove that the two triangles constructed from the segments P A, P B, P C and QA, QB, QC are of equal area.76. (VIE 5) Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter 6.25. Find the sides of the triangle.77. (YUG 1) By h(n), where n is an integer greater than 1, let us denote the greatest prime divisor of the number n. Are there infinitely many numbers n for which h(n) < h(n + 1) < h(n + 2) holds?78. (YUG 2) By ω(n), where n is an integer greater than 1, let us denote the number of different prime divisors of the number n. Prove that there
3.21 IMO 1979 139 exist infinitely many numbers n for which ω(n) < ω(n + 1) < ω(n + 2) holds.79. (YUG 3) Let S be a unit circle and K a subset of S consisting of several closed arcs. Let K satisfy the following properties: (i) K contains three points A, B, C, that are the vertices of an acute- angled triangle; (ii) for every point A that belongs to K its diametrically opposite point A and all points B on an arc of length 1/9 with center A do not belong to K. Prove that there are three points E, F, G on S that are vertices of an equilateral triangle and that do not belong to K.80. (YUG 4) (SL79-26).81. (YUG 5) Let P be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped P0 can be decomposed into parallelepipeds P1 , P2 , . . . , Pn ∈ P, prove that P0 ∈ P.3.21.3 Shortlisted Problems 1. (BEL 1) Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length). 2. (BEL 4) From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative? 3. (BUL 1) Find all polynomials f (x) with real coefficients for which f (x)f (2x2 ) = f (2x3 + x). 4. (BUL 3)IMO2 A pentagonal prism A1 A2 . . . A5 B1 B2 . . . B5 is given. The edges, the diagonals of the lateral walls 5. (CZS 2) Let n ≥ 2 be an integer. Find the maximal cardinality of a set M of pairs (j, k) of integers, 1 ≤ j < k ≤ n, with the following property: If (j, k) ∈ M , then (k, m) ∈ M for any m. 6. (CZS 4) Find the real values of p for which the equation 2p + 1 − x2 + 3x + p + 4 = x2 + 9x + 3p + 9
140 3 Problems √ in x has exactly two real distinct roots ( t means the positive square root of t). 7. (FRG 1)IMO1 Given that 1 − 1 + 1 − 1 + · · · − 1318 + 1319 = p , where 2 3 4 1 1 q p and q are natural numbers having no common factor, prove that p is divisible by 1979. 8. (FRG 3) For all rational x satisfying 0 ≤ x < 1, f is defined by f (2x)/4, for 0 ≤ x < 1/2, f (x) = 3/4 + f (2x − 1)/4, for 1/2 ≤ x < 1. Given that x = 0.b1 b2 b3 . . . is the binary representation of x, find f (x). 9. (FRG 4)IMO6 Let S and F be two opposite vertices of a regular octagon. A counter starts at S and each second is moved to one of the two neigh- boring 210. (FIN 3) Show that for any vectors a, b in Euclidean space, √ 3 3 2 2 |a × b| ≤ 3 |a| |b| |a − b|2 . 8 Remark. Here × denotes the vector product.11. (GDR 1) Given real numbers x1 , x2 , . . . , xn (n ≥ 2), with xi ≥ 1/n (i = 1, 2, . . . , n) and with x2 + x2 + · · · + x2 = 1, find whether the product 1 2 n P = x1 x2 x3 · · · xn has a greatest and/or least value and if so, give these values.12. (GDR 3) Let R be a set of exactly 6 elements. A set F of subsets of R is called an S-family over R if and only if it satisfies the following three conditions: (i) For no two sets X, Y in F is X ⊆ Y ; (ii) For any three sets X, Y, Z in F , X ∪ Y ∪ Z = R, (iii) X∈F X = R. We define |F | to be the number of elements of F (i.e., the number of subsets of R belonging to F ). Determine, if it exists, h = max |F |, the maximum being taken over all S-families over R.13. (GRE 1) Show that 20 60 < sin 20◦ < 21 60 .14. (GRE 5) Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.
3.21 IMO 1979 14115. (ISR 2)IMO5 The nonnegative real numbers x1 , x2 , x3 , x4 , x5 , a satisfy the following relations: 5 5 5 ixi = a, i3 xi = a2 , i5 xi = a3 . i=1 i=1 i=1 What are the possible values of a?16. (ISR 4) Let K denote the set {a, b, c, d, e}. F is a collection of 16 different subsets of K, and it is known that any three members of F have at least one element in common. Show that all 16 members of F have exactly one element in common.17. (NET 1) Inside an equilateral triangle ABC one constructs points P , Q and R such that ∠QAB = ∠P BA = 15◦ , ∠RBC = ∠QCB = 20◦ , ∠P CA = ∠RAC = 25◦ . Determine the angles of triangle P QR.18. (POL 1) Let m positive integers a1 , . . . , am be given. Prove that there exist fewer than 2m positive integers b1 , . . . , bn such that all sums of dis- tinct bk 's are distinct and all ai (i ≤ m) occur among them.19. (ROM 1) Consider the sequences (an ), (bn ) defined by a1 = 3, b1 = 100, an+1 = 3an , bn+1 = 100bn . Find the smallest integer m for which bm > a100 .20. (SWE 2) Given the integer n > 1 and the real number a > 0 determine the maximum of n−1 xi xi+1 taken over all nonnegative numbers xi with i=1 sum a.21. (USS 1) Let N be the number of integral solutions of the equation x2 − y 2 = z 3 − t3 satisfying the condition 0 ≤ x, y, z, t ≤ 106 , and let M be the number of integral solutions of the equation x2 − y 2 = z 3 − t3 + 1 satisfying the condition 0 ≤ x, y, z, t ≤ 106 . Prove that N > M .22. (USS 3)IMO3
142 3 Problems23. (USA 4) Find all natural numbers n for which 28 + 211 + 2n is a perfect square.24. (USA 5) A circle O with center O on base BC of an isosceles triangle ABC is tangent to the equal sides AB, AC. If point P on AB and point Q on AC are selected such that P B × CQ = (BC/2)2 , prove that line segment P Q is tangent to circle O, and prove the converse.25. (USA 6)IMO4 Given a point P in a given plane π and also a given point Q not in π, show how to determine a point R in π such that QP +P R is a QR maximum.26. (YUG 4) Prove that the functional equations f (x + y) = f (x) + f (y), and f (x + y + xy) = f (x) + f (y) + f (xy) (x, y ∈ R) are equivalent.
3.22 IMO 1981 1433.22 The Twenty-Second IMOWashington DC, United States of America, July 8–20,19813.22.1 Contest Problems First Day (July 13)1. Find the point P inside the triangle ABC for which BC CA AB + + PD PE PF is minimal, where P D, P E, P F are the perpendiculars from P to BC, CA, AB respectively.2. Let f (n, r) be the arithmetic mean of the minima of all r-subsets of the set {1, 2, . . . , n}. Prove that f (n, r) = n+1 . r+13. Determine the maximum value of m2 + n2 where m and n are integers satisfying m, n ∈ {1, 2, . . . , 1981} and (n2 − mn − m2 )2 = 1. Second Day (July 14)4. (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set in the set is a divisor of the least common multiple of the remaining n − 1 numbers? (b) For which values of n > 2 is there a unique set having the stated property?5. Three equal circles touch the sides of a triangle and have one common point O. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle ABC and the point O are collinear.6.4, 1981).
144 3 Problems3.22.2 Shortlisted Problems1. (BEL)IMO4 (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n − 1 numbers? (b) For which values of n > 2 is there a unique set having the stated property?2. (BUL) A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahe- dron ABCD at the points E, F, G, H respectively. The points E, F, G, H are the vertices of a square. Prove that if the sphere is tangent to the edge AC, then it is also tangent to the edge BD.3. (CAN) Find the minimum value of max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g) subject to the constraints (i) a, b, c, d, e, f, g ≥ 0, (ii) a + b + c + d + e + f + g = 1.4. (CAN) Let {fn } be the Fibonacci sequence {1, 1, 2, 3, 5, . . . }. (a) Find all pairs (a, b) of real numbers such that for each n, afn + bfn+1 is a member of the sequence. (b) Find all pairs (u, v) of positive real numbers such that for each n, 2 2 ufn + vfn+1 is a member of the sequence.5. (COL) A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.6. (CUB) Let P (z) and Q(z) be complex-variable polynomials, with degree not less than 1. Let Pk = {z ∈ C | P (z) = k}, Qk = {z ∈ C | Q(z) = k}. Let also P0 = Q0 and P1 = Q1 . Prove that P (z) ≡ Q(z).7. (FIN)IMO62, 2), f (3, 3) and f (4, 4). Alternative version: Determine f (4, 1981).
146 3 Problems18. (USS) Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.19. (YUG) A finite set of unit circles is given in a plane such that the area of their union U is S. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that 2S . 9
3.23 IMO 1982 1473.23 The Twenty-Third IMOBudapest, Hungary, July 5–14, 19823.23.1 Contest Problems First Day (July 9)1. The function f (n) is defined for all positive integers n and takes on non- negative integer values. Also, for all m, n, nonisosceles3. Consider the infinite sequences {xn } of positive real numbers with the following properties: x0 = 1 and for all i ≥ 0, xi+1 ≤ xi . (a) Prove that for every such sequence there is an n ≥ 1 such that x2 x2 x2 0 + 1 + · · · + n−1 ≥ 3.999. x1 x2 xn x2 x2 x2 (b) Find such a sequence for which 0 x1 + 1 x2 + ···+ n−1 xn < 4 for all n. Second Day (July 10)4. Prove that if n is a positive integer such that the equation x3 −3xy 2 +y 3 = n has a solution in integers (x, y), then it has at least three such solutions. Show that the equation has no solution in integers when n = 2891.5. The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N , respectively, so that AM = CN = r. AC CE Determine r if B, M , and N are collinear.6. that 2 the distance between X and Y is not greater than 1 and the length of the part of L that lies between X and Y is not smaller than 198.
148 3 Problems3.23.2 Longlisted Problems1. (AUS 1) It is well known that the binomial coefficients n = k!(n−k)! , k n! 0 ≤ k ≤ n, are positive integers. The factorial n! is defined inductively by 0! = 1, n! = n · (n − 1)! for n ≥ 1. (a) Prove that n+1 2n is an integer for n ≥ 0. 1 n (b) Given a positive integer k, determine the smallest integer Ck with the property that n+k+1 n+k is an integer for all n ≥ k. Ck 2n2. (AUS 2) Given a finite number of angular regions A1 , . . . , Ak in a plane, each Ai being bounded by two half-lines meeting at a vertex and provided with a + or − sign, we assign to each point P of the plane and not on a bounding half-line the number k − l, where k is the number of + regions and l the number of − regions that contain P . (Note that the boundary of Ai does not belong to Ai .) A For instance, in the figure we have ¡e + C ¡$ B e two + regions QAP and RCQ, and R$$$−e $ + ¡ one − region RBP . Every point in- ¡ Q eP side ABC receives the number ¡ e +1, while every point not inside ABC and not on a boundary halfline the number 0. We say that the interior of ABC is represented as a sum of the signed angular regions QAP , RBP , and RCQ. (a) Show how to represent the interior of any convex planar polygon as a sum of signed angular regions. (b) Show how to represent the interior of a tetrahedron as a sum of signed solid angular regions, that is, regions bounded by three planes inter- secting at a vertex and provided with a + or − sign.3. (AUS 3) Given n points X1 , X2 , . . . , Xn in the interval 0 ≤ Xi ≤ 1, i = 1, 2, . . . , n, show that there is a point y, 0 ≤ y ≤ 1, such that n 1 1 |y − Xi | = . n i=1 24. (AUS 4) (SL82-14). Original formulation. Let ABCD be a convex planar quadrilateral and let A1 denote the circumcenter of BCD. Define B1 , C1 , D1 in a corre- sponding circumcenter of B1 C1 D1 , and define B2 , C2 , D2 in an analogous way. Show that the quadrilateral A2 B2 C2 D2 is similar to the quadrilateral ABCD.
3.23 IMO 1982 149 (c) If the quadrilateral A1 B1 C1 D1 was obtained from the quadrilat- eral ABCD by the above process, what condition must be satis- fied by the four points A1 , B1 , C1 , D1 ? Assuming that the four points A1 , B1 , C1 , D1 satisfying this condition are given, describe a construc- tion by straightedge and compass to obtain the original quadrilateral ABCD. (It is not necessary to actually perform the construction). 5. (BEL 1) Among all triangles with a given perimeter, find the one with the maximal radius of its incircle. 6. (BEL 2) On the three distinct lines a, b, and c three points A, B, and C are given, respectively. Construct three collinear points X, Y, Z on lines BY CZ a, b, c, respectively, such that AX = 2 and AX = 3. 7. (BEL 3) Find all solutions (x, y) ∈ Z2 of the equation x3 − y 3 = 2xy + 8. 8. (BRA 1) (SL82-10). 9. (BRA 2) Let n be a natural number, n ≥ 2, and let φ be Euler's function; i.e., φ(n) is the number of positive integers not exceeding n and coprime to n. Given any two real numbers α and β, 0 ≤ α < β ≤ 1, prove that there exists a natural number m such that φ(m) α< < β. m10. (BRA 3) Let r1 , . . . , rn be the radii of n spheres. Call S1 , S2 , . . . , Sn the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that S1 S2 Sn 2 + r2 + · · · + r2 = 4π. r1 2 n11. (BRA 4) A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers a and b. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between a and b be for this to happen?12. (BRA 5) Let there be 3399 numbers arbitrarily chosen among the first 6798 integers 1, 2, . . . , 6798 in such a way that none of them divides an- other. Prove that there are exactly 1982 numbers in {1, 2, . . . , 6798} that must end up being chosen.13. (BUL 1) A regular n-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by S1 , S2 , and S, respectively. Let σ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that
150 3 Problems π σS = 4S1 S2 cos2 . n14. (BUL 2) (SL82-4).15. (CAN 1) Show that the set S of natural numbers n for which 3/n cannot be written as the sum of two reciprocals of natural numbers (S = {n | 3/n = 1/p + 1/q for any p, q ∈ N}) is not the union of finitely many arithmetic progressions.16. (CAN 2) (SL82-7).17. (CAN 3) (SL82-11).18. (CAN 4) You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that (a + ab−1 a)−1 + (a + b)−1 = a−1 , where x−1 is the element for which x−1 x = xx−1 = e, where e is the element of the system such that for all a the equality ea = ae = a holds.19. (CAN 5) (SL82-15).20. (CZS 1) Consider a cube C and two planes σ, τ , which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.21. (CZS 2) All edges and all diagonals of regular hexagon A1 A2 A3 A4 A5 A6 are colored blue or red such that each triangle Aj Ak Am , 1 ≤ j < k < m ≤ 6 has at least one red edge. Let Rk be the number of red segments Ak Aj , (j = k). Prove the inequality 6 (2Rk − 7)2 ≤ 54. k=122. (CZS 3) (SL82-19).23. (FIN 1) Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.24. (FIN 2) Prove that if a person a has infinitely many descendants (chil- dren, their children, etc.), then a has an infinite sequence a0 , a1 , . . . of descendants (i.e., a = a0 and for all n ≥ 1, an+1 is always a child of an ). It is assumed that no-one can have infinitely many children. Variant 1. Prove that if a has infinitely many ancestors, then a has an infinite descending sequence of ancestors (i.e., a0 , a1 , . . . where a = a0 and an is always a child of an+1 ). Variant 2. Prove that if someone has infinitely many ancestors, then all people cannot descend from A(dam) and E(ve).
152 3 Problems38. (POL 1) Numbers un,k (1 ≤ k ≤ n) are defined as follows: n u1,1 = 1, un,k = − un/d,k/d k d|n, d|k, d>1 (the empty sum is defined to be equal to zero). Prove that n | un,k for every natural number n and for every k (1 ≤ k ≤ n).39. (POL 2) Let S be the unit circle with center O and let P1 , P2 , . . . , Pn −→ − be points of S such that the sum of vectors vi = OPi is the zero vector. n Prove that the inequality i=1 XPi ≥ n holds for every point X.40. (POL 3) We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a 3k × n rectangle, then it is impossible to reach a position with only one pawn on the board.41. (POL 4) (SL82-8).42. (POL 5) Let F be the family of all k-element subsets of the set {1, 2, . . . , 2k + 1}. Prove that there exists a bijective function f : F → F such that for every A ∈ F, the sets A and f (A) are disjoint.43. (TUN 1) (a) What is the maximal number of acute angles in a convex polygon? (b) Consider m points in the interior of a convex n-gon. The n-gon is partitioned into triangles whose vertices are among the n + m given points (the vertices of the n-gon and the given points). Each of the m points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.44. (TUN 2) Let A and B be positions of two ships M and N , respectively, at the moment when N saw M moving with constant speed v following the line Ax. In search of help, N moves with speed kv (k < 1) along the line By in order to meet M as soon as possible. Denote by C the point of meeting of the two ships, and set π AB = d, ∠BAC = α, 0 ≤ α < . 2 Determine the angle ∠ABC = β and time t that N needs in order to meet M.45. (TUN 3) (SL82-20).46. (USA 1) Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.
154 3 Problems2 (YUG 1) Let K be a convex polygon in the plane and suppose that K is positioned in the coordinate system in such a way that 1 area (K ∩ Qi ) = area K (i = 1, 2, 3, 4, ), 4 where the Qi denote the quadrants of the plane. Prove that if K contains no nonzero lattice point, then the area of K is less than 4.3. A3 (USS 4)IMO3 Consider the infinite sequences {xn } of positive real numbers with the following properties: x0 = 1 and for all i ≥ 0, xi+1 ≤ xi . x2 (a) Prove that for every such sequence there is an n ≥ 1 such that 0 x1 + x2 x2 1 x2 + ···+ n−1 xn ≥ 3.999. x2 x2 x2 (b) Find such a sequence for which 0 x1 + 1 x2 + ···+ n−1 xn < 4 for all n.4. A4 (BUL 2) Determine all real values of the parameter a for which the equation 16x4 − ax3 + (2a + 17)x2 − ax + 16 = 0 has exactly four distinct real roots that form a geometric progression.5. A5 (NET 2)IMO5 Let A1 A2 A3 A4 A5 A6 be a regular hexagon. Each of its λ diagonals Ai−1 Ai+1 is divided into the same ratio 1−λ , where 0 < λ < 1, by a point Bi in such a way that Ai , Bi , and Bi+2 are collinear (i ≡ 1, . . . , 6 (mod 6)). Compute λ.6. A6 (VIE 1)IMO6 2 that the distance between X and Y is not greater than 1 and the length of that part of L that lies between X and Y is not smaller than 198.7. B1 (CAN 2) Let p(x) be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1)+p(−1)−2(1+p(0)).8. B2 (POL 4) A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.
3.23 IMO 1982 155 9. B3 (GBR 1) Let ABC be a triangle, and let P be a point inside it such that P AC = P BC. The perpendiculars from P to BC and CA meet these lines at L and M , respectively, and D is the midpoint of AB. Prove that DL = DM .10. B4 (BRA 1) A box contains p white balls and q black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?11. B5 (CAN 3) (a) Find the rearrangement {a1 , . . . , an } of {1, 2, . . . , n} that maximizes a1 a2 + a2 a3 + · · · + an a1 = Q. (b) Find the rearrangement that minimizes Q.12. B6 (FIN 3) Four distinct circles C, C1 , C2 , C3 and a line L are given in the plane such that C and L are disjoint and each of the circles C1 , C2 , C3 touches the other two, as well as C and L. Assuming the radius of C to be 1, determine the distance between its center and L.13. C1 (NET 1)IMO2 A scalene14. C2 (AUS 4) Let ABCD be a convex plane quadrilateral and let A1 denote the circumcenter of BCD. Define B1 , C1 , D1 in a corresponding circumcenter of B1 C1 D1 , and define B2 , C2 , D2 in an analogous way. Show that the quadrilateral A2 B2 C2 D2 is similar to the quadrilateral ABCD.15. C3 (CAN 5) Show that 1 − sa ≤ (1 + s)a−1 1−s holds for every 1 = s > 0 real and 0 < a ≤ 1 rational.
156 3 Problems16. C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the equation x3 − 3xy 2 + y 3 = n has a solution in integers (x, y), then it has at least three such solutions. Show that the equation has no solution in integers when n = 2891.17. C5 (USS 5) The right triangles ABC and AB1 C1 are similar and have opposite orientation. The right angles are at C and C1 , and we also have CAB = C1 AB1 . Let M be the point of intersection of the lines BC1 and B1 C. Prove that if the lines AM and CC1 exist, they are perpendic- ular.18. C6 (FRA 2) Let O be a point of three-dimensional space and let l1 , l2 , l3 be mutually perpendicular straight lines passing through O. Let S denote the sphere with center O and radius R, and for every point M of S, let SM denote the sphere with center M and radius R. We denote by P1 , P2 , P3 the intersection of SM with the straight lines l1 , l2 , l3 , respectively, where we put Pi = O if li meets SM at two distinct points and Pi = O otherwise (i = 1, 2, 3). What is the set of centers of gravity of the (possibly degenerate) triangles P1 P2 P3 as M runs through the points of S? m+n19. C7 (CZS 3) Let M be the set of real numbers of the form √m2 +n2 , where m and n are positive integers. Prove that for every pair x ∈ M , y ∈ M with x < y, there exists an element z ∈ M such that x < z < y.20. C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri- angles ABM , CDP , BCN , ADQ, the first two outward and the other two inward. Prove that M N = AC. What can be said about the quadrilateral M N P Q?
3.24 IMO 1983 1573.24 The Twenty-Fourth IMOParis, France, July 1–12, 19833.24.1 Contest Problems First Day (July 6)1. Find all functions f defined on the positive real numbers and taking pos- itive real values that satisfy the following conditions: (i) f (xf (y)) = yf (x) for all positive real x, y; (ii) f (x) → 0 as x → +∞.2. Let K be one of the two intersection points of the circles W1 and W2 . Let O1 and O2 be the centers of W1 and W2 . The two common tangents to the circles meet W1 and W2 respectively in P1 and P2 , the first tangent, and Q1 and Q2 the second tangent. Let M1 and M2 be the midpoints of P1 Q1 and P2 Q2 , respectively. Prove that ∠O1 KO2 = ∠M1 KM2 .3. Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) = 1. Show that 2abc − ab − bc − ca is the largest integer not representable as xbc + yca + zab with nonnegative integers x, y, z. Second Day (July 7)4. Let ABC be an equilateral triangle. Let E be the set of all points from segments AB, BC, and CA (including A, B, and C). Is it true that for any partition of the set E into two disjoint subsets, there exists a right-angled triangle all of whose vertices belong to the same subset in the partition?5. Prove or disprove the following statement: In the set {1, 2, 3, . . . , 105 } a subset of 1983 elements can be found that does not contain any three consecutive terms of an arithmetic progression.6. If a, b, and c are sides of a triangle, prove that a2 b(a − b) + b2 c(b − c) + c2 a(c − a) ≥ 0 and determine when there is equality.3.24.2 Longlisted Problems1. (AUS 1) (SL83-1).2. (AUS 2) Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.
158 3 Problems 3. (AUS 3) (a) Given a tetrahedron ABCD and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from D passes through the orthocenter H4 of ∆ABC. Prove that this altitude DH4 intersects all the other three altitudes. (b) If we further know that a second altitude, say the one from vertex A to the face BCD, also passes through the orthocenter H1 of ∆BCD, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle. 4. (BEL 1) (SL83-2). 5. (BEL 2) Consider the set Q2 of points in R2 , both of whose coordinates are rational. (a) Prove that the union of segments with vertices from Q2 is the entire set R2 . (b) Is the convex hull of Q2 (i.e., the smallest convex set in R2 that con- tains Q2 ) equal to R2 ? 6. (BEL 3) (SL83-3). 7. (BEL 4) Find all numbers x ∈ Z for which the number x4 + x3 + x2 + x + 1 is a perfect square. 8. (BEL 5) (SL83-4). 9. (BRA 1) (SL83-5).10. (BRA 2) Which of the numbers 1, 2, . . . , 1983 has the largest number of divisors?11. (BRA 3) A boy at point A wants to get water at a circular lake and carry it to point B. Find the point C on the lake such that the distance walked by the boy is the shortest possible given that the line AB and the lake are exterior to each other.12. (BRA 4) The number 0 or 1 is to be assigned to each of the n vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?13. (BUL 1) Let p be a prime number and a1 , a2 , . . . , a(p+1)/2 different nat- ural numbers less than or equal to p. Prove that for each natural number r less than or equal to p, there exist two numbers (perhaps equal) ai and aj such that p ≡ ai aj (mod r).
3.24 IMO 1983 15914. (BUL 2) Let l be tangent to the circle k at B. Let A be a point on k and P the foot of perpendicular from A to l. Let M be symmetric to P with respect to AB. Find the set of all such points M .15. (CAN 1) Find all possible finite sequences {n0 , n1 , n2 , . . . , nk } of integers such that for each i, i appears in the sequence ni times (0 ≤ i ≤ k).16. (CAN 2) (SL83-6).17. (CAN 3) In how many ways can 1, 2, . . . , 2n be arranged in a 2 × n a1 a2 · · · a n rectangular array for which: b1 b2 · · · bn (i) a1 < a2 < · · · < an , (ii) b1 < b2 < · · · < bn , (iii) a1 < b1 , a2 < b2 , . . . , an < bn ?18. (CAN 4) Let b ≥ 2 be a positive integer. (a) Show that for an integer N , written in base b, to be equal to the sum of the squares of its digits, it is necessary either that N = 1 or that N have only two digits. (b) Give a complete list of all integers not exceeding 50 that, relative to some base b, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base b there is an integer other than 1 that is equal to the sum of the squares of its digits.19. (CAN 5) (SL83-7).20. (COL 1) Let f and g be functions from the set A to the same set A. We define f to be a functional nth root of g (n is a positive integer) if f n (x) = g(x), where f n (x) = f n−1 (f (x)). (a) Prove that the function g : R → R, g(x) = 1/x has an infinite number of nth functional roots for each positive integer n. (b) Prove that there is a bijection from R onto R that has no nth func- tional root for each positive integer n.21. (COL 2) Prove that there are infinitely many positive integers n for which it is possible for a knight, starting at one of the squares of an n × n chessboard, to go through each of the squares exactly once.22. (CUB 1) Does there exist an infinite number of sets C consisting of 1983 consecutive natural numbers such that each of the numbers is divisible by some number of the form a1983 , with a ∈ N, a = 1?23. (FIN 1) (SL83-10).24. (FIN 2) Every x, 0 ≤ x ≤ 1, admits a unique representation x = ∞ −j j=0 aj 2 , where all the aj belong to {0, 1} and infinitely many of them 1+c 1 are 0. If b(0) = 2+c , b(1) = 2+c , c > 0, and
160 3 Problems ∞ f (x) = a0 + b(a0 ) · · · b(aj )aj+1 , j=0 show that 0 < f (x) − x < c for every x, 0 < x < 1. (FIN 2 ) (SL83-11).25. (FRG 1) How many permutations a1 , a2 , . . . , an of {1, 2, . . . , n} are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange ai and ai+1 whenever ai > ai+1 for i running from 1 up to n − 1?26. (FRG 2) Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) = 1. Show that 2abc − ab − bc − ca cannot be represented as bcx + cay + abz with nonnegative integers x, y, z.27. (FRG 3) (SL83-18).28. (GBR 1) Show that if the sides a, b, c of a triangle satisfy the equation 2(ab2 + bc2 + ca2 ) = a2 b + b2 c + c2 a + 3abc, then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.29. (GBR 2) Let O be a point outside a given circle. Two lines OAB, OCD through O meet the circle at A, B, C, D, where A, C are the midpoints of OB, OD, respectively. Additionally, the acute angle θ between the lines is equal to the acute angle at which each line cuts the circle. Find cos θ and show that the tangents at A, D to the circle meet on the line BC.30. (GBR 3) Prove the existence of a unique sequence {un } (n = 0, 1, 2 . . . ) of positive integers such that n n+r u2 = n un−r for all n ≥ 0, r=0 r m where r is the usual binomial coefficient.31. (GBR 4) (SL83-12).32. (GBR 5) Let a, b, c be positive real numbers and let [x] denote the greatest integer that does not exceed the real number x. Suppose that f is a function defined on the set of nonnegative integers n and taking real values such that f (0) = 0 and f (n) ≤ an + f ([bn]) + f ([cn]), for all n ≥ 1. Prove that if b + c < 1, there is a real number k such that f (n) ≤ kn for all n, (1)
3.24 IMO 1983 161 while if b + c = 1, there is a real number K such that f (n) ≤ Kn log2 n for all n ≥ 2. Show that if b + c = 1, there may not be a real number k that satisfies (1).33. (GDR 1) (SL83-16).34. (GDR 2) In a plane are given n points Pi (i = 1, 2, . . . , n) and two angles α and β. Over each of the segments Pi Pi=1 (Pn+1 = P1 ) a point Qi is constructed such that for all i: (i) upon moving from Pi to Pi+1 , Qi is seen on the same side of Pi Pi+1 , (ii) ∠Pi+1 Pi Qi = α, (iii) ∠Pi Pi+1 Qi = β. Furthermore, let g be a line in the same plane with the property that all the points Pi , Qi lie on the same side of g. Prove that n n d(Pi , g) = d(Qi , g), i=1 i=1 where d(M, g) denotes the distance from point M to line g.35. (GDR 3) (SL83-17).36. (ISR 1) The set X has 1983 members. There exists a family of subsets {S1 , S2 , . . . , Sk } such that: (i) the union of any three of these subsets is the entire set X, while (ii) the union of any two of them contains at most 1979 members. What is the largest possible value of k?37. (ISR 2) The points A1 , A2 , . . . , A1983 are set on the circumference of a circle and each is given one of the values ±1. Show that if the number of points with the value +1 is greater than 1789, then at least 1207 of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.38. (KUW 1) Let {un } be the sequence defined by its first two terms u0 , u1 and the recursion formula un+2 = un − un+1 . (a) Show that un can be written in the form un = αan + βbn , where a, b, α, β are constants independent of n that have to be determined. (b) If Sn = u0 + u1 + · · · + un , prove that Sn + un−1 is a constant inde- pendent of n. Determine this constant.39. (KUW 2) If α is the real root of the equation E(x) = x3 − 5x − 50 = 0 such that xn+1 = (5xn + 50)1/3 and x1 = 5, where n is a positive integer, prove that:
162 3 Problems (a) x3 − α3 = 5(xn − α) n+1 (b) α < xn+1 < xn40. (LUX 1) Four faces of tetrahedron ABCD are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are a < b < c, determine the radius of the sphere circumscribed about the tetrahedron as a function on a, b, and c. What is the ratio c/a if R = a?41. (LUX 2) (SL83-13).42. (LUX 3) Consider the square ABCD in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square ABCD.43. (LUX 4) Given a square ABCD, let P , Q, R, and S be four variable points on the sides AB, BC, CD, and DA, respectively. Determine the positions of the points P , Q, R, and S for which the quadrilateral P QRS is a parallelogram, a rectangle, a square, or a trapezoid.44. (LUX 5) We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.45. (LUX 6) Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass 2, and then take the same spoon of the new mixture from glass 2 and pour it back into the first glass. What happens after this operation is repeated n times, and what as n tends to infinity?46. (LUX 7) Let f be a real-valued function defined on I = (0, +∞) and having no zeros on I. Suppose that f (x) lim = +∞. x→+∞ f (x) f (n+1) For the sequence un = ln f (n) , prove that un → +∞ (n → +∞).47. (NET 1) In a plane, three pairwise intersecting circles C1 , C2 , C3 with centers M1 , M2 , M3 are given. For i = 1, 2, 3, let Ai be one of the points of intersection of Cj and Ck ({i, j, k} = {1, 2, 3}). Prove that if ∠M3 A1 M2 = ∠M1 A2 M3 = ∠M2 A3 M1 = π/3 (directed angles), then M1 A1 , M2 A2 , and M3 A3 are concurrent.48. (NET 2) Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
164 3 Problems61. (SWE 2) Let a and b be integers. Is it possible to find integers p and q such that the integers p + na and q + nb have no common prime factor no matter how the integer n is chosen.62. (SWE 3) A circle γ is drawn and let AB be a diameter. The point C on γ is the midpoint of the line segment BD. The line segments AC and DO, where O is the center of γ, intersect at P . Prove that there is a point E on AB such that P is on the circle with diameter AE.63. (SWE 4) (SL83-22).64. (USA 1) The sum of all the face angles about all of the vertices except one of a given polyhedron is 5160. Find the sum of all of the face angles of the polyhedron.65. (USA 2) Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect in a point P . Prove that AP cot ∠BAC + cot ∠DAC = . PC cot ∠BCA + cot ∠DCA66. (USA 3) (SL83-9).67. (USA 4) The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.68. (USA 5) Three of the roots of the equation x4 − px3 + qx2 − rx + s = 0 are tan A, tan B, and tan C, where A, B, and C are angles of a triangle. Determine the fourth root as a function only of p, q, r, and s.69. (USS 1) (SL83-23).70. (USS 2) (SL83-24).71. (USS 3) (SL83-25).72. (USS 4) Prove that for all x1 , x2 , . . . , xn ∈ R the following inequality holds: n(n − 2) cos2 (xi − xj ) ≥ . 4 n≥i>j≥173. (VIE 1) Let ABC be a nonequilateral triangle. Prove that there exist two points P and Q in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of ABC, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.74. (VIE 2) In a plane we are given two distinct points A, B and two lines a, b passing through B and A respectively (a B, b A) such that the line AB is equally inclined to a and b. Find the locus of points M in the plane such that the product of distances from M to A and a equals the
3.24 IMO 1983 165 product of distances from M to B and b (i.e., M A · M A = M B · M B , where A and B are the feet of the perpendiculars from M to a and b respectively).75. (VIE 3) Find the sum of the fiftieth powers of all sides and diagonals of a regular 100-gon inscribed in a circle of radius R.3.24.3 Shortlisted Problems 1. (AUS 1) The localities P1 , P2 , . . . , P1983 are served by ten international airlines A1 , A2 , . . . , A10 . It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings. 2. (BEL 1) Let n be a positive integer. Let σ(n) be the sum of the natural divisors d of n (including 1 and n). We say that an integer m ≥ 1 is superabundant (P.Erd¨s, 1944) if ∀k ∈ {1, 2, . . . , m − 1}, σ(m) > σ(k) . o m k Prove that there exists an infinity of superabundant numbers. 3. (BEL 3)IMO4 We say that a set E of points of the Euclidian plane is "Pythagorean" if for any partition of E into two sets A and B, at least one of the sets contains the vertices of a right-angled triangle. Decide whether the following sets are Pythagorean: (a) a circle; (b) an equilateral triangle (that is, the set of three vertices and the points of the three edges). 4. (BEL 5) On the sides of the triangle ABC, three similar isosceles tri- angles ABP (AP = P B), AQC (AQ = QC), and BRC (BR = RC) are constructed. The first two are constructed externally to the triangle ABC, but the third is placed in the same half-plane determined by the line BC as the triangle ABC. Prove that AP RQ is a parallelogram. 5. (BRA 1) Consider the set of all strictly decreasing sequences of n natural numbers having the property that in each sequence no term divides any other term of the sequence. Let A = (aj ) and B = (bj ) be any two such sequences. We say that A precedes B if for some k, ak < bk and ai = bi for i < k. Find the terms of the first sequence of the set under this ordering. 6. (CAN 2) Suppose that {x1 , x2 , . . . , xn } are positive integers for which x1 + x2 + · · · + xn = 2(n + 1). Show that there exists an integer r with 0 ≤ r ≤ n − 1 for which the following n − 1 inequalities hold: xr+1 + · · · + xr+i ≤ 2i + 1 ∀i, 1 ≤ i ≤ n − r; xr+1 + · · · + xn + x1 + · · · + xi ≤ 2(n − r + i) + 1 ∀i, 1 ≤ i ≤ r − 1. Prove that if all the inequalities are strict, then r is unique and that otherwise there are exactly two such r.
166 3 Problems 7. (CAN 5) Let a be a positive integer and let {an } be defined by a0 = 0 and an+1 = (an + 1)a + (a + 1)an + 2 a(a + 1)an (an + 1) (n = 1, 2 . . . ). Show that for each positive integer n, an is a positive integer. 8. (SPA 2) In a test, 3n students participate, who are located in three rows of n students in each. The students leave the test room one by one. If N1 (t), N2 (t), N3 (t) denote the numbers of students in the first, second, and third row respectively at time t, find the probability that for each t during the test, |Ni (t) − Nj (t)| < 2, i = j, i, j = 1, 2, . . . . 9. (USA 3)IMO6 If a, b, and c are sides of a triangle, prove that a2 b(a − b) + b2 c(b − c) + c2 a(c − a) ≥ 0. Determine when there is equality.10. (FIN 1) Let p and q be integers. Show that there exists an interval I of length 1/q and a polynomial P with integral coefficients such that p 1 P (x) − < 2 q q for all x ∈ I.11. (FIN 2 ) Let f : [0, 1] → R be continuous and satisfy: bf (2x) = f (x), 0 ≤ x ≤ 1/2; f (x) = b + (1 − b)f (2x − 1), 1/2 ≤ x ≤ 1, where b = 1+c 2+c , c > 0. Show that 0 < f (x) − x < c for every x, 0 < x < 1.12. (GBR 4)IMO1 Find all functions f defined on the positive real numbers and taking positive real values that satisfy the following conditions: (i) f (xf (y)) = yf (x) for all positive real x, y. (ii) f (x) → 0 as x → +∞.13. (LUX 2) Let E be the set of 19833 points of the space R3 all three of whose coordinates are integers between 0 and 1982 (including 0 and 1982). A coloring of E is a map from E to the set {red, blue}. How many colorings of E are there satisfying the following property: The number of red vertices among the 8 vertices of any right-angled parallelepiped is a multiple of 4?14. (POL 2)IMO5 Prove or disprove: From the interval [1, . . . , 30000] one can select a set of 1000 integers containing no arithmetic triple (three consecutive numbers of an arithmetic progression).
168 3 Problems24. (USS 2) Let dn be the last nonzero digit of the decimal representation of n!. Prove that dn is aperiodic; that is, there do not exist T and n0 such that for all n ≥ n0 , dn+T = dn .25. (USS 3) Prove that every partition of 3-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every a ∈ R+ , there are points M and N inside that subset such that distance between M and N is exactly a.
3.25 IMO 1984 1693.25 The Twenty-Fifth IMOPrague, Czechoslovakia, June 29–July 10, 19843.25.1 Contest Problems First Day (July 4)1. Let x, y, z be nonnegative real numbers with x + y + z = 1. Show that 7 0 ≤ xy + yz + zx − 2xyz ≤ . 272. Find two positive integers a, b such that none of the numbers a, b, a + b is divisible by 7 and (a + b)7 − a7 − b7 is divisible by 77 .3. In a plane two different points O and A are given. For each point X = O of the plane denote by α(X) the angle AOX measured in radians (0 ≤ α(X) < 2π) and by C(X) the circle with center O and radius OX + α(X) . OX Suppose each point of the plane is colored by one of a finite number of colors. Show that there exists a point X with α(X) > 0 such that its color appears somewhere on the circle C(X). Second Day (July 5)4. Let ABCD be a convex quadrilateral for which the circle of diameter AB is tangent to the line CD. Show that the circle of diameter CD is tangent to the line AB if and only if the lines BC and AD are parallel.5. Let d be the sum of the lengths of all diagonals of a convex polygon of n (n > 3) vertices, and let p be its perimeter. Prove that n−3 d 1 n n+1 < < −2 . 2 p 2 2 26. Let a, b, c, d be odd positive integers such that a < b < c < d, ad = bc, and a + d = 2k , b + c = 2m for some integers k and m. Prove that a = 1.3.25.2 Longlisted Problems 31. (AUS 1) The fraction 10 can be written as the sum of two positive fractions with numerator 1 as follows: 10 = 1 + 10 and also 10 = 1 + 20 . 3 5 1 3 4 1 There are the only two ways in which this can be done. 3 In how many ways can 1984 be written as the sum of two positive fractions with numerator 1? 3 Is there a positive integer n, not divisible by 3, such that n can be written as the sum of two positive fractions with numerator 1 in exactly 1984 ways?
3.25 IMO 1984 17110. (BUL 4) Assume that the bisecting plane of the dihedral angle at edge AB of the tetrahedron ABCD meets the edge CD at point E. Denote by S1 , S2 , S3 , respectively the areas of the triangles ABC, ABE, and ABD. Prove that no tetrahedron exists for which S1 , S2 , S3 (in this order) form an arithmetic or geometric progression.11. (BUL 5) (SL84-13).12. (CAN 1) (SL84-11). Original formulation. Suppose that a1 , a2 , . . . , a2n are distinct integers such that (x − a1 )(x − a2 ) · · · (x − a2n ) + (−1)n−1 (n!)2 = 0 a1 +a2 +···+a2n has an integer solution r. Show that r = 2n .13. (CAN 2) (SL84-2). Original formulation. Let m, n be nonzero integers. Show that 4mn−m−n can be a square infinitely many times, but that this never happens when either m or n is positive. Alternative formulation. Let m, n be positive integers. Show that 4mn − m − n can be 1 less than a perfect square infinitely often, but can never be a square.14. (CAN 3) (SL84-6).15. (CAN 4) Consider all the sums of the form 1985 ek k 5 = ±15 ± 25 ± · · · ± 19855, k=1 where ek = ±1. What is the smallest nonnegative value attained by a sum of this type?16. (CAN 5) (SL84-19).17. (FRA 1) (SL84-1).18. (FRA 2) Let c be the inscribed circle of the triangle ABC, d a line tan- gent to c which does not pass through the vertices of triangle ABC. Prove the existence of points A1 , B1 , C1 , respectively, on the lines BC, CA, AB satisfying the following two properties: (i) Lines AA1 , BB1 , and CC1 are parallel. (ii) Lines AA1 , BB1 , and CC1 meet d respectively at points A , B , and C such that A A1 B B1 C C1 = = . AA BB CC19. (FRA 3) Let ABC be an isosceles triangle with right angle at point A. Find the minimum of the function F given by √ F (M ) = BM + CM − 3AM.
172 3 Problems20. (FRG 1) (SL84-5).21. (FRG 2) (1) Start with a white balls and b black balls. (2) Draw one ball at random. (3) If the ball is white, then stop. Otherwise, add two black balls and go to step 2. Let S be the number of draws before the process terminates. For the cases a = b = 1 and a = b = 2 only, find an = P (S = n), bn = P (S ≤ n), limn→∞ bn , and the expectation value of the number of balls drawn: E(S) = n≥1 nan .22. (FRG 3) (SL84-17). Original formulation. In a permutation (x1 , x2 , . . . , xn ) of the set 1, 2, . . . , n we call a pair (xi , xj ) discordant if i < j and xi > xj . Let d(n, k) be the number of such permutations with exactly k discordant pairs. (a) Find d(n, 2). (b) Show that d(n, k) = d(n, k − 1) + d(n − 1, k) − d(n − 1, k − 1) with d(n, k) = 0 for k < 0 and d(n, 0) = 1 for n ≥ 1. Compute with this recursion a table of d(n, k) for n = 1 to 6.23. (FRG 4) A 2 × 2 × 12 box fixed in space is to be filled with twenty-four 1 × 1 × 2 bricks. In how many ways can this be done?24. (FRG 5) (SL84-7). Original formulation. Consider several types of 4-cell figures: (a) (b) (c) (d) (e) . Find, with proof, for which of these types of figures it is not possible to number the fields of the 8 × 8 chessboard using the numbers 1, 2, . . . , 64 in such a way that the sum of the four numbers in each of its parts congruent to the given figure is divisible by 4.25. (GBR 1) (SL84-10).26. (GBR 2) A cylindrical container has height 6 cm and radius 4 cm. It rests on a circular hoop, also of radius 4 cm, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder's vertical ends.27. (GBR 3) The function f (n) is defined on the nonnegative integers n by: f (0) = 0, f (1) = 1,
174 3 Problems36. (MON 4) The set {1, 2, . . . , 49} is divided into three subsets. Prove that at least one of these subsets contains three different numbers a, b, c such that a + b = c.37. (MOR 1) Denote by [x] the greatest integer not exceeding x. For all real k > 1, define two sequences: nk an (k) = [nk] and bn (k) = . k−1 If A(k) = {an (k) : n ∈ N} and B(k) = {bn (k) : n ∈ N}, prove that A(k) and B(k) form a partition of N if and only if k is irrational.38. (MOR 2) Determine all continuous functions f such that 2 ∀(x, y) ∈ R2 f (x + y)f (x − y) = (f (x)f (y)) .39. (MOR 3) Let ABC be an isosceles triangle, AB = AC, ∠A = 20◦ . Let D be a point on AB, and E a point on AC such that ∠ACD = 20◦ and ∠ABE = 30◦ . What is the measure of the angle ∠CDE?40. (NET 1) (SL84-12).41. (NET 2) Determine positive integers p, q, and r such that the diagonal of a block consisting of p × q × r unit cubes passes through exactly 1984 of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.) 142. (NET 3) Triangle ABC is given for which BC = AC + 2 AB. The point P divides AB such that RP : P A = 1 : 3. Prove that ∠CAP = 2∠CP A.43. (POL 1) (SL84-16).44. (POL 2) (SL84-9).45. (POL 3) Let X be an arbitrary nonempty set contained in the plane and let sets A1 , A2 , . . . , Am and B1 , B2 , . . . , Bn be its images under parallel translations. Let us suppose that A1 ∪ A2 ∪ · · · ∪ Am ⊂ B1 ∪ B2 ∪ · · · ∪ Bn and that the sets A1 , A2 , . . . , Am are disjoint. Prove that m ≤ n.46. (ROM 1) Let (an )n≥1 and (bn )n≥1 be two sequences of natural numbers such that an+1 = nan + 1, bn+1 = nbn − 1 for every n ≥ 1. Show that these two sequences can have only a finite number of terms in common.47. (ROM 2) (SL84-8).48. (ROM 3) Let ABC be a triangle with interior angle bisectors AA1 , 1 BB1 , CC1 and incenter I. If σ[IA1 B] + σ[IB1 C] + σ[IC1 A] = 2 σ[ABC], where σ[ABC] denotes the area of ABC, show that ABC is isosceles.
3.25 IMO 1984 17549. (ROM 4) Let n > 1 and xi ∈ R for i = 1, . . . , n. Set Sk = xk + xk + 1 2 · · · + xk for k ≥ 1. If S1 = S2 = · · · = Sn+1 , show that xi ∈ {0, 1} for n every i = 1, 2, . . . , n.50. (ROM 5) (SL84-14).51. (SPA 1) Two cyclists leave simultaneously a point P in a circular run- way with constant velocities v1 , v2 (v1 > v2 ) and in the same sense. A +v pedestrian leaves P at the same time, moving with velocity v3 = v112 2 . If the pedestrian and the cyclists move in opposite directions, the pedes- trian meets the second cyclist 91 seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him 187 seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.52. (SPA 2) Construct a scalene triangle such that a(tan B − tan C) = b(tan A − tan C). i+453. (SPA 3) Find a sequence of natural numbers ai such that ai = r=1 dr , where dr = ds for r = s and dr divides ai .54. (SPA 4) Let P be a convex planar polygon with equal angles. Let l1 , . . . , ln be its sides. Show that a necessary and sufficient condition for li P to be regular is that the sum of the ratios li+1 (i = 1, . . . , n; ln+1 = l1 ) equals the number of sides.55. (SPA 5) Let a, b, c be natural numbers such that a+b+c = 2pq(p30 +q 30 ), p > q being two given positive integers. (a) Prove that k = a3 + b3 + c3 is not a prime number. (b) Prove that if a · b · c is maximum, then 1984 divides k.56. (SWE 1) Let a, b, c be nonnegative integers such that a ≤ b ≤ c, 2b = a + c and a+b+c is an integer. Is it possible to find three nonnegative 3 integers d, e, and f such that d ≤ e ≤ f , f = c, and such that a2 +b2 +c2 = d2 + e2 + f 2 ?57. (SWE 2) Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let n = (10a + b)10c+d . Find the probability that 1984! is divisible by n.58. (SWE 3) Let (an )∞ be a sequence such that an ≤ an+m ≤ an + am for 1 all positive integers n and m. Prove that an has a limit as n approaches n infinity.59. (USA 1) Determine the smallest positive integer m such that 529n + m · 132n is divisible by 262417 for all odd positive integers n.60. (USA 2) (SL84-20).61. (USA 3) A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.
176 3 Problems62. (USA 4) From a point P exterior to a circle K, two rays are drawn intersecting K in the respective pairs of points A, A and B, B . For any other pair of points C, C on K, let D be the point of intersection of the circumcircles of triangles P AC and P B C other than point P . Similarly, let D be the point of intersection of the circumcircles of triangles P A C and P BC other than point P . Prove that the points P , D, and D are collinear.63. (USA 5) (SL84-18).64. (USS 1) For a matrix (pij ) of the format m × n with real entries, set n m ai = pij for i = 1, . . . , m and bj = pij for j = 1, . . . , n. (1) j=1 i=1 By integering a real number we mean replacing the number with the in- teger closest to it. Prove that integering the numbers ai , bj , pij can be done in such a way that (1) still holds.65. (USS 2) A tetrahedron is inscribed in a sphere of radius 1 such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.66. (USS 3) (SL84-3). Original formulation. All the divisors of a positive integer n arranged in increasing order are x1 < x2 < · · · < xk . Find all such numbers n for which x2 + x2 − 1 = n. 5 667. (USS 4) With the medians of an acute-angled triangle another triangle is constructed. If R and Rm are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that 5 Rm > R. 668. (USS 5) In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher "Martian Words" makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume. Find the word whose numeration is equal to the sum of numerations of the words Prague, Olympiad, Mathematics.3.25.3 Shortlisted Problems 1. (FRA 1) Find all solutions of the following system of n equations in n variables:
180 3 Problems3.26 The Twenty-Sixth IMOJoutsa, Finland, June 29–July 11, 19853.26.1 Contest Problems First Day (July 4)1. A circle whose center is on the side ED of the cyclic quadrilateral BCDE touches the other three sides. Prove that EB + CD = ED.2. relatively prime to n, i and |j − i| receive the same color for all i ∈ N , i = j. Prove that all numbers in N must receive the same color.3. The weight w(p) of a polynomial p, p(x) = n ai xi , with integer coeffi- i=0 cients the inequality w(qi1 + · · · + qin ) ≥ w(qi1 ) holds. Second Day (July 5)4. Given a set M of 1985 positive integers, none of which has a prime divisor larger than 26, prove that M has four distinct elements whose geometric mean is an integer.5. OM B = 90◦ .6.3.26.2 Longlisted Problems1. (AUS 1) (SL85-4).
3.26 IMO 1985 181 2. (AUS 2) We are given a triangle ABC and three rectangles R1 , R2 , R3 with sides parallel to two fixed perpendicular directions and such that their union covers the sides AB, BC, and CA; i.e., each point on the perimeter of ABC is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of R1 , R2 , R3 . 3. (AUS 3) A function f has the following property: If k > 1, j > 1, and (k, j) = m, then f (kj) = f (m) (f (k/m) + f (j/m)). What values can f (1984) and f (1985) take? 4. (BEL 1) Let x, y, and z be real numbers satisfying x + y + z = xyz. Prove that x(1 − y 2 )(1 − z 2 ) + y(1 − z 2 )(1 − x2 ) + z(1 − x2 )(1 − y 2 ) = 4xyz. 5. (BEL 2) (SL85-16). 6. (BEL 3) On a one-way street, an unending sequence of cars of width a, length b passes with velocity v. The cars are separated by the distance c. A pedestrian crosses the street perpendicularly with velocity w, without paying attention to the cars. (a) What is the probability that the pedestrian crosses the street unin- jured? (b) Can he improve this probability by crossing the road in a direction other than perpendicular? 7. (BRA 1) A convex quadrilateral is inscribed in a circle of radius 1. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than 2. 8. (BRA 2) Let K be a convex set in the xy-plane, symmetric with respect to the origin and having area greater than 4. Prove that there exists a point (m, n) = (0, 0) in K such that m and n are integers. 9. (BRA 3) (SL85-2).10. (BUL 1) (SL85-13).11. (BUL 2) Let a and b be integers and n a positive integer. Prove that bn−1 a(a + b)(a + 2b) · · · (a + (n − 1)b) n! is an integer.12. (CAN 1) Find the maximum value of sin2 θ1 + sin2 θ2 + · · · + sin2 θn subject to the restrictions 0 ≤ θi ≤ π, θ1 + θ2 + · · · + θn = π.13. (CAN 2) Find the average of the quantity
3.26 IMO 1985 183 S(r, 0) ∩ S(u, v) = ∅, S(r, 0) ∪ S(u, v) = N.24. (FRA 1) Let d ≥ 1 be an integer that is not the square of an integer. Prove that for every integer n ≥ 1, √ √ (n d + 1)| sin(nπ d)| ≥ 1.25. (FRA 2) Find eight positive integers n1 , n2 , . . . , n8 with the follow- ing property: For every integer k, −1985 ≤ k ≤ 1985, there are eight integers α1 , α2 , . . . , α8 , each belonging to the set {−1, 0, 1}, such that 8 k = i=1 αi ni .26. (FRA 3) (SL85-15).27. (FRA 4) Let O be a point on the oriented Euclidean plane and (i, j) a directly oriented orthonormal basis. Let C be the circle of radius 1, centered at O. For every real number t and nonnegative integer n let Mn be −− −→ −− −→ the point on C for which i, OMn = cos 2n t (or OMn = cos 2n ti+sin 2n tj). Let k ≥ 2 be an integer. Find all real numbers t ∈ [0, 2π) that satisfy (i) M0 = Mk , and (ii) if one starts from M0 and goes once around C in the positive direction, one meets successively the points M0 , M1 , . . . , Mk−2 , Mk−1 , in this order.28. (FRG 1) Let M be the set of the lengths of an octahedron whose sides are congruent quadrangles. Prove that M has at most three elements. (FRG 1a) Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.29. (FRG 2) Call a four-digit number (xyzt)B in the number system with base B stable if (xyzt)B = (dcba)B − (abcd)B , where a ≤ b ≤ c ≤ d are the digits of (xyzt)B in ascending order. Determine all stable numbers in the number system with base B. (FRG 2a) The same problem with B = 1985. (FRG 2b) With assumptions as in FRG 2, determine the number of bases B ≤ 1985 such that there is a stable number with base B.30. (GBR 1) A plane rectangular grid is given and a "rational point" is defined as a point (x, y) where x and y are both rational numbers. Let A, B, A , B be four distinct rational points. Let P be a point such that AB B P PA AB = BC = P A . In other words, the triangles ABP , A B P are directly or oppositely similar. Prove that P is in general a rational point and find the exceptional positions of A and B relative to A and B such that there exists a P that is not a rational point.31. (GBR 2) Let E1 , E2 , and E3 be three mutually intersecting ellipses, all in the same plane. Their foci are respectively F2 , F3 ; F3 , F1 ; and F1 , F2 . The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.
184 3 Problems32. (GBR 3) A collection of 2n letters contains 2 each of n different letters. The collection is partitioned into n pairs, each pair containing 2 letters, which may be the same or different. Denote the number of distinct parti- tions by un . (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are not considered distinct.) Prove that un+1 = (n + 1)un − n(n−1) un−2 . 2 (GBR 3a) A pack of n cards contains n pairs of 2 identical cards. It is shuffled and 2 cards are dealt to each of n different players. Let pn be the probability that every one of the n players is dealt two identical cards. Prove that pn+1 = n+1 − n(n−1) . 1 pn 2pn−233. (GBR 4) (SL85-12).34. (GBR 5) (SL85-20).35. (GDR 1) We call a coloring f of the elements in the set M = {(x, y) | x = 0, 1, . . . , kn − 1; y = 0, 1, . . . , ln − 1} with n colors allowable if every color appears exactly k and l times in each row and column and there are no rectangles with sides parallel to the coordinate axes such that all the vertices in M have the same color. Prove that every allowable coloring f satisfies kl ≤ n(n + 1).36. (GDR 2) Determine whether there exist 100 distinct lines in the plane having exactly 1985 distinct points of intersection.37. (GDR 3) Prove that a triangle with angles α, β, γ, circumradius R, and area A satisfies α β γ 9R2 tan + tan + tan ≤ . 2 2 2 4A38. (IRE 1) (SL85-21).39. (IRE 2) Given a triangle ABC and external points X, Y , and Z such that BAZ = CAY , CBX = ABZ, and ACY = BCX, prove that AX, BY , and CZ are concurrent.40. (IRE 3) Each of the numbers x1 , x2 , . . . , xn equals 1 or −1 and x1 x2 x3 x4 + x2 x3 x4 x5 + · · · + xn−3 xn−2 xn−1 xn +xn−2 xn−1 xn x1 + xn−1 xn x1 x2 + xn x1 x2 x3 = 0. Prove that n is divisible by 4.41. (IRE 4) (SL85-14).42. (ISR 1) Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.43. (ISR 2) Suppose that 1985 points are given inside a unit cube. Show that one can always choose 32 of them in such a way that every (possibly
3.26 IMO 1985 185 degenerate) closed polygon with these points as vertices has a total length √ of less than 8 3.44. (ISR 3) (SL85-19).45. (ITA 1) Two persons, X and Y , play with a die. X wins a game if the outcome is 1 or 2; Y wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within 5 games. Determine the probabilities of winning in an unlimited number of games. If X bets 1, how much must Y bet for the game to be fair?46. (ITA 2) Let C be the curve determined by the equation y = x3 in the rectangular coordinate system. Let t be the tangent to C at a point P of C; t intersects C at another point Q. Find the equation of the set L of the midpoints M of P Q as P describes C. Is the correspondence associating P and M a bijection of C on L? Find a similarity that transforms C into L.47. (ITA 3) Let F be the correspondence associating with every point P = (x, y) the point P = (x , y ) such that x = ax + b, y = ay + 2b. (1) Show that if a = 1, all lines P P are concurrent. Find the equation of the set of points corresponding to P = (1, 1) for b = a2 . Show that the composition of two mappings of type (1) is of the same type.48. (ITA 4) In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, A, B, and C. Person A says, "If C is a knight, B is a knave." Person C says, "A and I are different; one is a knight and the other is a knave." Who are the knights, and who are the knaves?49. (MON 1) (SL85-1).50. (MON 2) From each of the vertices of a regular n-gon a car starts to move with constant speed along the perimeter of the n-gon in the same direction. Prove that if all the cars end up at a vertex A at the same time, then they never again meet at any other vertex of the n-gon. Can they meet again at A?51. (MON 3) Let f1 = (a1 , a2 , . . . , an ), n > 2, be a sequence of integers. From f1 one constructs a sequence fk of sequences as follows: if fk = (c1 , c2 , . . . , cn ), then fk+1 = (ci1 , ci2 , ci3 + 1, ci4 + 1, . . . , cin + 1), where (ci1 , ci2 , . . . , cin ) is a permutation of (c1 , c2 , . . . , cn ). Give a necessary and sufficient condition for f1 under which it is possible for fk to be a constant sequence (b1 , b2 , . . . , bn ), b1 = b2 = · · · = bn , for some k.52. (MON 4) In the triangle ABC, let B1 be on AC, E on AB, G on BC, and let EG be parallel to AC. Furthermore, let EG be tangent to the
186 3 Problems inscribed circle of the triangle ABB1 and intersect BB1 at F . Let r, r1 , and r2 be the inradii of the triangles ABC, ABB1 , and BF G, respectively. Prove that r = r1 + r2 .53. (MON 5) For each P inside the triangle ABC, let A(P ), B(P ), and C(P ) be the points of intersection of the lines AP , BP , and CP with the sides opposite to A, B, and C, respectively. Determine P in such a way that the area of the triangle A(P )B(P )C(P ) is as large as possible. n 554. (MOR 1) Set Sn = p=1 (p + p7 ). Determine the greatest common divisor of Sn and S3n .55. (MOR 2) The points A, B, C are in this order on line D, and AB = 4BC. Let M be a variable point on the perpendicular to D through C. Let M T1 and M T2 be tangents to the circle with center A and radius AB. Determine the locus of the orthocenter of the triangle M T1 T2 .56. (MOR 3) Let ABCD be a rhombus with angle ∠A = 60◦ . Let E be a point, different from D, on the line AD. The lines CE and AB intersect at F . The lines DF and BE intersect at M . Determine the angle BM D as a function of the position of E on AD.57. (NET 1) The solid S is defined as the intersection of the six spheres with the six edges of a regular tetrahedron T , with edge length 1, as diameters. 1 Prove that S contains two points at a distance √6 . (NET 1a) Using the same assumptions, prove that no pair of points in 1 S has a distance larger than √6 .58. (NET 2) Prove that there are infinitely many pairs (k, N ) of positive integers such that 1 + 2 + · · · + k = (k + 1) + (k + 2) + · · · + N .59. (NET 3) (SL85-3). n60. (NOR 1) The sequence (sn ), where sn = k=1 sin k, n = 1, 2, . . . , is bounded. Find an upper and lower bound.61. (NOR 2) Consider the set A = {0, 1, 2, . . . , 9} and let (B1 , B2 , . . . , Bk ) be a collection of nonempty subsets of A such that Bi ∩ Bj has at most two elements for i = j. What is the maximal value of k?62. (NOR 3) A "large" circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed π cm 3 per second (relative to the disk). Describe the path of the insect (a) relative to the disk; (b) relative to the wall.63. (POL 1) (SL85-6).64. (POL 2) Let p be a prime. For which k can the set {1, 2, . . . , k} be partitioned into p subsets with equal sums of elements?
3.26 IMO 1985 18765. (POL 3) Define the functions f, F : N → N, by √ 3− 5 f (n) = n , F (k) = min{n ∈ N|f k (n) > 0}, 2 where f k = f ◦ · · · ◦ f is f iterated n times. Prove that F (k + 2) = 3F (k + 1) − F (k) for all k ∈ N.66. (ROM 1) (SL85-5).67. (ROM 2) Let k ≥ 2 and n1 , n2 , . . . , nk ≥ 1 natural numbers having the property n2 | 2n1 − 1, n3 | 2n2 − 1, . . . , nk | 2nk−1 − 1, and n1 | 2nk − 1. Show that n1 = n2 = · · · = nk = 1. √68. (ROM 3) Show that the sequence {an }n≥1 defined by an = [n 2] contains an infinite number of integer powers of 2. ([x] is the integer part of x.)69. (ROM 4) Let A and B be two finite disjoint sets of points in the plane such that any three distinct points in A∪B are not collinear. Assume that at least one of the sets A, B contains at least five points. Show that there exists a triangle all of whose vertices are contained in A or in B that does not contain in its interior any point from the other set.70. (ROM 5) Let C be a class of functions √ : N → N that contains the f functions S(x) = x + 1 and E(x) = x − [ x]2 for every x ∈ N. ([x] is the integer part of x.) If C has the property that for every f, g ∈ C, f + g, f g, f ◦ g ∈ C, show that the function max(f (x) − g(x), 0) is in C.71. (ROM 6) For every integer r > 1 find the smallest integer h(r) > 1 having the following property: For any partition of the set {1, 2, . . . , h(r)} into r classes, there exist integers a ≥ 0, 1 ≤ x ≤ y such that the numbers a + x, a + y, a + x + y are contained in the same class of the partition.72. (SPA 1) Construct a triangle ABC given the side AB and the distance OH from the circumcenter O to the orthocenter H, assuming that OH and AB are parallel.73. (SPA 2) Let A1 A2 , B1 B2 , C1 C2 be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines B2 C1 , C2 A1 , A2 B1 , the segments B2 C1 , C2 A1 , A2 B1 are proportional to the sides in which they are contained.74. (SPA 3) Find the triples of positive integers x, y, z satisfying 1 1 1 4 + + = . x y z 575. (SPA 4) Let ABCD be a rectangle, AB = a, BC = b. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being d) that are at an the angle φ, 0 ≤ φ ≤ 90◦ ,
188 3 Problems with respect to AB. Let L be the sum of the lengths of all the segments intersecting the rectangle. Find: (a) how L varies, (b) a necessary and sufficient condition for L to be a constant, and (c) the value of this constant.76. (SWE 1) Are there integers m and n such that 5m2 − 6mn + 7n2 = 1985?77. (SWE 2) Two equilateral triangles are inscribed in a circle with radius r. Let A be the area of the set consisting of all points interior to both √ triangles. Prove that 2A ≥ r2 3.78. (SWE 3) (SL85-17).79. (SWE 4) Let a, b, and c be real numbers such that 1 1 1 + + = 0. bc − a2 ca − b2 ab − c2 Prove that a b c + + = 0. (bc − a2 )2 (ca − b2 )2 (ab − c2 )280. (TUR 1) Let E = {1, 2, . . . , 16} and let M be the collection of all 4 × 4 matrices whose entries are distinct members of E. If a matrix A = (aij )4×4 is chosen randomly from M , compute the probability p(k) of maxi minj aij = k for k ∈ E. Furthermore, determine l ∈ E such that p(l) = max{p(k) | k ∈ E}.81. (TUR 2) Given the side a and the corresponding altitude ha of a triangle ABC, find a relation between a and ha such that it is possible to construct, with straightedge and compass, triangle ABC such that the altitudes of ABC form a right triangle admitting ha as hypotenuse.82. (TUR 3) Find all cubic polynomials x3 + ax2 + bx + c admitting the rational numbers a, b, and c as roots.83. (TUR 4) Let Γi , i = 0, 1, 2, . . . , be a circle of radius ri inscribed in an angle of measure 2α such that each Γi is externally tangent to Γi+1 and ri+1 < ri . Show that the sum of the areas of the circles Γi is equal to the √ √ 1 area of a circle of radius r = 2 r0 ( sin α + csc α).84. (TUR 5) (SL85-8).85. (USA 1) Let CD be a diameter of circle K. Let AB be a chord that is parallel to CD. The line segment AE, with E on K, is parallel to CB; F is the point of intersection of line segments AB and DE. The line segment F G, with G on DC, extended is parallel to CB. Is GA tangent to K at point A?
3.26 IMO 1985 18986. (USA 2) Let l denote the length of the smallest diagonal of all rectangles inscribed in a triangle T . (By inscribed, we mean that all four vertices of the rectangle lie on the boundary of T .) Determine the maximum value l2 of S(T ) taken over all triangles (S(T ) denotes the area of triangle T ).87. (USA 3) (SL85-9).88. (USA 4) Determine the range of w(w + x)(w + y)(w + z), where x, y, z, and w are real numbers such that x + y + z + w = x7 + y 7 + z 7 + w7 = 0.89. (USA 5) Given that n elements a1 , a2 , . . . , an are organized into n pairs P1 , P2 , . . . , Pn in such a way that two pairs Pi , Pj share exactly one el- ement when (ai , aj ) is one of the pairs, prove that every element is in exactly two of the pairs.90. (USS 1) Decompose the number 51985 −1 into a product of three integers, each of which is larger than 5100 .91. (USS 2) Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the answers she got were different. How many people did the deputy leader of the Illyrian team greet?92. (USS 3) (SL85-11). (USS 3a) Given six numbers, find a method of computing by using not more than 15 additions and 14 multiplications the following five numbers: the sum of the numbers, the sum of products of the numbers taken two at a time, and the sums of the products of the numbers taken three, four, and five at a time.93. (USS 4) The sphere inscribed in tetrahedron ABCD touches the sides ABD and DBC at points K and M , respectively. Prove that AKB = DM C.94. (USS 5) (SL85-22).95. (VIE 1) (SL85-10). (VIE 1a) Prove that for each point M on the edges of a regular tetrahe- dron there is one and only one point M on the surface of the tetrahedron such that there are at least three curves joining M and M on the sur- face of the tetrahedron of minimal length among all curves joining M and M on the surface of the tetrahedron. Denote this minimal length by dM . Determine the positions of M for which dM attains an extremum.96. (VIE 2) Determine all functions f : R → R satisfying the following two conditions:
190 3 Problems (a) f (x + y) + f (x − y) = 2f (x)f (y) for all x, y ∈ R, (b) limx→∞ f (x) = 0.97. (VIE 3) In a plane a circle with radius R and center w and a line Λ are given. The distance between w and Λ is d, d > R. The points M and N are chosen on Λ in such a way that the circle with diameter M N is externally tangent to the given circle. Show that there exists a point A in the plane such that all the segments M N are seen in a constant angle from A.3.26.3 Shortlisted ProblemsProposals of the Problem Selection Committee. 1. (MON 1)IMO4 Given a set M of 1985 positive integers, none of which has a prime divisor larger than 26, prove that the set has four distinct elements whose geometric mean is an integer. 2. (BRA 3) A polyhedron has 12 faces and is such that: (i) all faces are isosceles triangles, (ii) all edges have length either x or y, (iii) at each vertex either 3 or 6 edges meet, and (iv) all dihedral angles are equal. Find the ratio x/y. n 3. (NET 3)IMO3 The weight w(p) of a polynomial p, p(x) = i=0 ai xi , with integer coefficients , the inequality w(qi1 + · · · + qin ) ≥ w(qi1 ) holds. 4. (AUS 1)IMO2 , relatively prime to n, i and |j − i| receive the same color for all i ∈ N , i = j. Prove that all numbers in N must receive the same color. 5. (ROM 1) Let D be the interior of the circle C and let A ∈ C. Show |MA| that the function f : D → R, f (M ) = |MM | , where M = (AM ∩ C, is strictly convex; i.e., f (P ) < f (M1 )+f (M2 ) , ∀M1 , M2 ∈ D, M1 = M2 , where 2 P is the midpoint of the segment M1 M2 . √ 6. (POL 1) Let xn = 2 2 + 3 3 + . . . + n n. Prove that
3.26 IMO 1985 191 1 xn+1 − xn < , n = 2, 3, . . . . n! Alternatives 7. 1a.(CZS 3) The positive integers x1 , . . . , xn , n ≥ 3, satisfy x1 < x2 < · · · < xn < 2x1 . Set P = x1 x2 · · · xn . Prove that if p is a prime number, k a positive integer, and P is divisible by pk , then pk ≥ n!. P 8. 1b.(TUR 5) Find the smallest positive integer n such that (i) n has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of n. 9. 2a.(USA 3) Determine the radius of a sphere S that passes through the centroids of each face of a given tetrahedron T inscribed in a unit sphere with center O. Also, determine the distance from O to the center of S as a function of the edges of T .10. 2b.(VIE 1) Prove that for every point M on the surface of a regular tetrahedron there exists a point M such that there are at least three different curves on the surface joining M to M with the smallest possible length among all curves on the surface joining M to M .11. 3a.(USS 3) Find a method by which one can compute the coefficients of P (x) = x6 + a1 x5 + · · · + a6 from the roots of P (x) = 0 by performing not more than 15 additions and 15 multiplications.12. 3b.(GBR 4) A sequence of polynomials Pm (x, y, z), m = 0, 1, 2, . . . , in x, y, and z is defined by P0 (x, y, z) = 1 and by Pm (x, y, z) = (x + z)(y + z)Pm−1 (x, y, z + 1) − z 2 Pm−1 (x, y, z) for m > 0. Prove that each Pm (x, y, z) is symmetric, in other words, is unaltered by any permutation of x, y, z.13. 4a.(BUL 1) Let m boxes be given, with some balls in each box. Let n < m be a given integer. The following operation is performed: choose n of the boxes and put 1 ball in each of them. Prove: (a) If m and n are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. (b) If m and n are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.14. 4b.(IRE 4) A set of 1985 points is distributed around the circumference of a circle and each of the points is marked with 1 or −1. A point is called "good" if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are
192 3 Problems all strictly positive. Show that if the number of points marked with −1 is less than 662, there must be at least one good point.15. 5a.(FRA 3) Let K and K be two squares in the same plane, their sides of equal length. Is it possible to decompose K into a finite number of tri- angles T1 , T2 , . . . , Tp with mutually disjoint interiors and find translations t1 , t2 , . . . , tp such that p K = ti (Ti )? i=116. 5b.(BEL 2) If possible, construct an equilateral triangle whose three vertices are on three given circles.17. 6a.(SWE 3)IMO618. 6b.(CAN 5) Let x1 , x2 , . . . , xn be positive numbers. Prove that x2 x2 x2 x2 1 + 2 2 + · · · + 2 n−1 + 2 n ≤ n − 1. x2 1 + x2 x3 x2 + x3 x4 xn−1 + xn x1 xn + x1 x2 Supplementary Problems19. (ISR 3) For which integers n ≥ 3 does there exist a regular n-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?20. (GBR 5)IMO1 A circle whose center is on the side ED of the cyclic quadrilateral BCDE touches the other three sides. Prove that EB+CD = ED.21. (IRE 1) The tangents at B and C to the circumcircle of the acute-angled triangle ABC meet at X. Let M be the midpoint of BC. Prove that (a) ∠BAM = ∠CAX, and (b) AM = cos ∠BAC. AX22. (USS 5)IMO5 ∠OM B = 90◦ .
3.27 IMO 1986 1933.27 The Twenty-Seventh IMOWarsaw, Poland, July 4–15, 19863.27.1 Contest Problems First Day (July 9)1.2. Let A, B, C be fixed points in the plane. A man starts from a certain point P0 and walks directly to A. At A he turns his direction by 60◦ to the left and walks to P1 such that P0 A = AP1 . After he performs the same action 1986 times successively around the points A, B, C, A, B, C, . . . , he returns to the starting point. Prove that ABC is an equilateral triangle, and that the vertices A, B, C are arranged counterclockwise.3. Second Day (July 10)4.5.6. Prove or disprove: Given a finite set of points with integer coefficients
194 3 Problems3.27.2 Longlisted Problems 1. (AUS 1) Let k be one of the integers 2, 3, 4 and let n = 2k − 1. Prove the inequality k 1 + bk + b2k + · · · + bnk ≥ (1 + bn ) for all real b ≥ 0. 2. (AUS 2) Let ABCD be a convex quadrilateral. DA and CB meet at F and AB and DC meet at E. The bisectors of the angles DF C and AED are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines AC and BD. 3. (AUS 3) A line parallel to the side BC of a triangle ABC meets AB in F and AC in E. Prove that the circles on BE and CF as diameters intersect in a point lying on the altitude of the triangle ABC dropped from A to BC. 4. (BEL 1) Find the last eight digits of the binary development of 271986 . 5. (BEL 2) Let ABC and DEF be acute-angled triangles. Write d = EF , e = F D, f = DE. Show that there exists a point P in the interior of ABC for which the value of the expression d · AP + e · BP + f · CP attains a minimum. 6. (BEL 3) In an urn there are one ball marked 1, two balls marked 2, and so on, up to n balls marked n. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number. 7. (BUL 1) (SL86-11). 8. (BUL 2) (SL86-19). 9. (CAN 1) In a triangle ABC, ∠BAC = 100◦, AB = AC. A point D is chosen on the side AC such that ∠ABD = ∠CBD. Prove that AD + DB = BC.10. (CAN 2) A set of n standard dice are shaken and randomly placed in a straight line. If n < 2r and r < s, then the probability that there will be a string of at least r, but not more than s, consecutive 1's can be written as P/6s+2 . Find an explicit expression for P .11. (CAN 3) (SL86-20).12. (CHN 1) Let O be an interior point of a tetrahedron A1 A2 A3 A4 . Let S1 , S2 , S3 , S4 be spheres with centers A1 , A2 , A3 , A4 , respectively, and let U, V be spheres with centers at O. Suppose that for i, j = 1, 2, 3, 4, i = j, the spheres Si and Sj are tangent to each other at a point Bij lying on Ai Aj . Suppose also that U is tangent to all edges Ai Aj and V is tangent to the spheres S1 , S2 , S3 , S4 . Prove that A1 A2 A3 A4 is a regular tetrahedron.
196 3 Problems22. (FRA 2) Let (an )n∈N be the sequence of integers defined recursively by a0 = 0, a1 = 1, an+2 = 4an+1 + an for n ≥ 0. Find the common divisors of a1986 and a6891 .23. (FRA 3) Let I and J be the centers of the incircle and the excircle in the angle BAC of the triangle ABC. For any point M in the plane of the triangle, not on the line BC, denote by IM and JM the centers of the incircle and the excircle (touching BC) of the triangle BCM . Find the locus of points M for which IIM JJM is a rectangle.24. (FRA 4) Two families of parallel lines are given in the plane, consisting of 15 and 11 lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let V be the set of 165 intersection points of the lines under consideration. Show that there exist not fewer than 1986 distinct squares with vertices in the set V .25. (FRA 5) (SL86-7).26. (FRG 1) (SL86-5).27. (FRG 2) In an urn there are n balls numbered 1, 2, . . . , n. They are drawn at random one by one one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.28. (FRG 3) (SL86-13).29. (FRG 4) We define a binary operation in the plane as follows: Given two points A and B in the plane, C = A B is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points I, M , O in the plane if I (M O) = (O I) M holds?30. (FRG 5) Prove that a convex polyhedron all of whose faces are equilat- eral triangles has at most 30 edges.31. (GBR 1) Let P and Q be distinct points in the plane of a triangle ABC such that AP : AQ = BP : BQ = CP : CQ. Prove that the line P Q passes through the circumcenter of the triangle.32. (GBR 2) Find, with proof, all solutions of the equation 1 x + 2 y − 3 z =1 in positive integers x, y, z.33. (GBR 3) (SL86-1).34. (GBR 4) For each nonnegative integer n, Fn (x) is a polynomial in x of degreee n. Prove that if the identity n n r Fn (2x) = (−1)n−r 2 Fr (x) r=0 r
3.27 IMO 1986 197 holds for each n, then n n r Fn (tx) = t (1 − t)n−r Fr (x) r=0 r for each n and all t.35. (GBR 5) Establish the maximum and minimum values that the sum |a| + |b| + |c| can have if a, b, c are real numbers such that the maximum value of |ax2 + bx + c| is 1 for −1 ≤ x ≤ 1.36. (GDR 1) (SL86-9).37. (GDR 2) Prove that the set {1, 2, . . . , 1986} can be partitioned into 27 disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).38. (GDR 3) (SL86-12).39. (GRE 1) Let S be a k-element set. (a) Find the number of mappings f : S → S such that (i) f (x) = x for x ∈ S, (ii) f (f (x)) = x for x ∈ S. (b) The same with the condition (i) left out.40. (GRE 2) Find the maximum value that the quantity 2m + 7n can have such that there exist distinct positive integers xi (1 ≤ i ≤ m), yj (1 ≤ j ≤ m n n) such that the xi 's are even, the yj 's are odd, and i=1 xi + j=1 yj = 1986.41. (GRE 3) Let M, N, P be the midpoints of the sides BC, CA, AB of a triangle ABC. The lines AM , BN , CP intersect the circumcircle of ABC at points A , B , C , respectively. Show that if A B C is an equilateral triangle, then so is ABC.42. (HUN 1) The integers 1, 2, . . . , n2 are placed on the fields of an n × n chessboard (n > 2) in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most n + 1. What is the total number of such placements?43. (HUN 2) (SL86-10).44. (IRE 1) (SL86-14).45. (IRE 2) Given n real numbers a1 ≤ a2 ≤ · · · ≤ an , define n 1 2 M1 = ai , M 2 = ai aj , Q = M1 − M2 . 2 n i=1 n(n − 1) 1≤i<j≤n Prove that a1 ≤ M 1 − Q ≤ M 1 + Q ≤ an and that equality holds if and only if a1 = a2 = · · · = an .
198 3 Problems46. (IRE 3) We wish to construct a matrix with 19 rows and 86 columns, with entries xij ∈ {0, 1, 2} (1 ≤ i ≤ 19, 1 ≤ j ≤ 86), such that: (i) in each column there are exactly k terms equal to 0; (ii) for any distinct j, k ∈ {1, . . . , 86} there is i ∈ {1, . . . , 19} with xij + xik = 3. For what values of k is this possible?47. (ISR 1) (SL86-16).48. (ISR 2) Let P be a convex 1986-gon in the plane. Let A, D be interior points of two distinct sides of P and let B, C be two distinct interior points of the line segment AD. Starting with an arbitrary point Q1 on the boundary of P , define recursively a sequence of points Qn as follows: given Qn extend the directed line segment Qn B to meet the boundary of P in a point Rn and then extend Rn C to meet the boundary of P again in a point, which is defined to be Qn+1 . Prove that for all n large enough the points Qn are on one of the sides of P containing A or D.49. (ISR 3) Let C1 , C2 be circles of radius 1/2 tangent to each other and both tangent internally to a circle C of radius 1. The circles C1 and C2 are the first two terms of an infinite sequence of distinct circles Cn defined as follows: Cn+2 is tangent externally to Cn and Cn+1 and internally to C. Show that the radius of each Cn is the reciprocal of an integer.50. (LUX 1) Let D be the point on the side BC of the triangle ABC such that AD is the bisector of ∠CAB. Let I be the incenter of ABC. (a) Construct the points P and Q on the sides AB and AC, respectively, such that P Q is parallel to BC and the perimeter of the triangle AP Q is equal to k · BC, where k is a given rational number. (b) Let R be the intersection point of P Q and AD. For what value of k does the equality AR = RI hold? (c) In which case do the equalities AR = RI = ID hold?51. (MON 1) Let a, b, c, d be the lengths of the sides of a quadrilateral √ circumscribed about a circle and let S be its area. Prove that S ≤ abcd and find conditions for equality.52. (MON 2) Solve the system of equations tan x1 + cot x1 = 3 tan x2 , tan x2 + cot x2 = 3 tan x3 , ··· ··· tan xn + cot xn = 3 tan x1 .53. (MON 3) For given positive integers r, v, n let S(r, v, n) denote the num- ber of n-tuples of nonnegative integers (x1 , . . . , xn ) satisfying the equation x1 + · · · + xn = r and such that xi ≤ v for i = 1, . . . , n. Prove that m n r − (v + 1)k + n − 1 S(r, v, n) = (−1)k , k n−1 k=0
200 3 Problems65. (ROM 5) Let A1 A2 A3 A4 be a quadrilateral inscribed in a circle C. Show that there is a point M on C such that M A1 − M A2 + M A3 − M A4 = 0.66. (SWE 1) One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.67. (SWE 2) (SL86-2).68. (SWE 3) Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real coefficients a, b. Determine the number of distinct real roots and their multiplicities for various values of a and b. Display your result graphically in the (a, b) plane.69. (TUR 1) (SL86-18).70. (TUR 2) (SL86-21).71. (TUR 3) Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.72. (TUR 4) A one-person game with two possible outcomes is played as follows: After each play, the player receives either a or b points, where a and b are integers with 0 < b < a < 1986. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer x ≥ 1986 can be obtained as the total score whereas 1985 and 663 cannot. Determine a and b.73. (TUR 5) Let (ai )i∈N be a strictly increasing sequence of positive real numbers such that limi→∞ ai = +∞ and ai+1 /ai ≤ 10 for each i. Prove that for every positive integer k there are infinitely many pairs (i, j) with 10k ≤ ai /aj ≤ 10k+1 .74. (USA 1) (SL86-8). Alternative formulation. Let A be a set of n points in space. From the fam- ily of all segments with endpoints in A, q segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of m yellow segments, where m ≥ 2q , arranged in order of increasing length. n75. (USA 2) The incenter of a triangle is the midpoint of the line seg- ment of length 4 joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.76. (USA 3) (SL86-3).77. (USS 1) Find all integers x, y, z that satisfy x3 + y 3 + z 3 = x + y + z = 8.
3.27 IMO 1986 20178. (USS 2) If T and T1 are two triangles with angles x, y, z and x1 , y1 , z1 , respectively, prove the inequality cos x1 cos y1 cos z1 + + ≤ cot x + cot y + cot z. sin x sin y sin z79. (USS 3) Let AA1 , BB1 , CC1 be the altitudes in an acute-angled triangle ABC. K and M are points on the line segments A1 C1 and B1 C1 respec- tively. Prove that if the angles M AK and CAA1 are equal, then the angle C1 KM is bisected by AK.80. (USS 4) Let ABCD be a tetrahedron and O its incenter, and let the line OD be perpendicular to AD. Find the angle between the planes DOB and DOC.3.27.3 Shortlisted Problems 1. (GBR 3)IMO5 2. (SWE 2) Let f (x) = xn where n is a fixed positive integer and x = 1, 2, . . . . Is the decimal expansion a = 0.f (1)f (2)f (3) . . . rational for any value of n? The decimal expansion of a is defined as follows: If f (x) = d1 (x)d2 (x) . . . . . . dr(x) (x) is the decimal expansion of f (x), then a = 0.1d1 (2)d2 (2) . . . . . . dr(2) (2)d1 (3) . . . dr(3) (3)d1 (4) . . . . 3. (USA 3) Let A, B, and C be three points on the edge of a circular chord such that B is due west of C and ABC is an equilateral triangle whose side is 86 meters long. A boy swam from A directly toward B. After covering a distance of x meters, he turned and swam westward, reaching the shore after covering a distance of y meters. If x and y are both positive integers, determine y. 4. (CZS 3) Let n be a positive integer and let p be a prime number, p > 3. Find at least 3(n+1) [easier version: 2(n+1)] sequences of positive integers x, y, z satisfying xyz = pn (x + y + z) that do not differ only by permutation. 5. (FRG 1)IMO1 6. (NET 1) Find four positive integers each not exceeding 70000 and each having more than 100 divisors.
202 3 Problems 7. (FRA 5) Let real numbers x1 , x2 , . . . , xn satisfy 0 < x1 < x2 < · · · < xn < 1 and set x0 = 0, xn+1 = 1. Suppose that these numbers satisfy the following system of equations: n+1 1 =0 where i = 1, 2, . . . , n. (1) xi − xj j=0, j=i Prove that xn+1−i = 1 − xi for i = 1, 2, . . . , n. 8. (USA 1) From a collection of n persons q distinct two-member teams are selected and ranked 1, . . . , q (no ties). Let m be the least integer larger than or equal to 2q/n. Show that there are m distinct teams that may be listed so that (i) each pair of consecutive teams on the list have one member in common and (ii) the chain of teams on the list are in rank order. Alternative formulation. Given a graph with n vertices and q edges num- bered 1, . . . , q, show that there exists a chain of m edges, m ≥ 2q , each n two consecutive edges having a common vertex, arranged monotonically with respect to the numbering. 9. (GDR 1)IMO6 Prove or disprove: Given a finite set of points with integer coordinates10. (HUN 2) Three persons A, B, C, are playing the following game: A k- element subset of the set {1, . . . , 1986} is randomly chosen, with an equal probability of each choice, where k is a fixed positive integer less than or equal to 1986. The winner is A, B or C, respectively, if the sum of the chosen numbers leaves a remainder of 0, 1, or 2 when divided by 3. For what values of k is this game a fair one? (A game is fair if the three outcomes are equally probable.)11. (BUL 1) Let f (n) be the least number of distinct points in the plane such that for each k = 1, 2, . . . , n there exists a straight line containing exactly k of these points. Find an explicit expression for f (n). Simplified version. Show that f (n) = n+1 n+2 ([x] denoting the great- 2 2 est integer not exceeding x).12. (GDR 3)IMO3
3.27 IMO 1986 20313. (FRG 3) A particle moves from (0, 0) to (n, n) directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At (n, y), y < n, it stays there if a head comes up and at (x, n), x < n, it stays there if a tail comes up. Let k be a fixed positive integer. Find the probability that the particle needs exactly 2n + k tosses to reach (n, n).14. (IRE 1) The circle inscribed in a triangle ABC touches the sides BC, CA, AB in D, E, F , respectively, and X, Y, Z are the midpoints of EF, F D, DE, respectively. Prove that the centers of the inscribed circle and of the circles around XY Z and ABC are collinear.15. (NET 2) Let ABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A B C D be a quadrangle such that A , B , C , D are the centers of the circumcircles of triangles BCD, ACD, ABD, and ABC. We write T (ABCD) = A B C D . Let us define A B C D = T (A B C D ) = T (T (ABCD)). (a) Prove that ABCD and A B C D are similar. (b) The ratio of similitude depends on the size of the angles of ABCD. Determine this ratio.16. (ISR 1)IMO417. (CHN 3)IMO2 Let A, B, C be fixed points in the plane. A man starts from a certain point P0 and walks directly to A. At A he turns his di- rection by 60◦ to the left and walks to P1 such that P0 A = AP1 . Af- ter he does the same action 1986 times successively around the points A, B, C, A, B, C, . . . , he returns to the starting point. Prove that ABC is equilateral and that the vertices A, B, C are arranged counterclockwise.18. (TUR 1) Let AX, BY, CZ be three cevians concurrent at an inte- rior point D of a triangle ABC. Prove that if two of the quadrangles DY AZ, DZBX, DXCY are circumscribable, so is the third.19. (BUL 2) A tetrahedron ABCD is given such that AD = BC = a; AC = BD = b; AB · CD = c2 . Let f (P ) = AP + BP + CP + DP , where P is an arbitrary point in space. Compute the least value of f (P ).20. (CAN 3) Prove that the sum of the face angles at each vertex of a tetra- hedron is a straight angle if and only if the faces are congruent triangles.21. (TUR 2) Let ABCD be a tetrahedron having each sum of opposite sides equal to 1. Prove that √ 3 rA + rB + rC + rD ≤ , 3 where rA , rB , rC , rD are the inradii of the faces, equality holding only if ABCD is regular.
204 3 Problems3.28 The Twenty-Eighth IMOHavana, Cuba, July 5–16, 19873.28.1 Contest Problems First Day (July 10)1. Let S be a set of n elements. We denote the number of all permutations of S that have exactly k fixed points by pn (k). Prove that n kpn (k) = n!. k=02. The prolongation of the bisector AL (L ∈ BC) in the acute-angled trian- gle ABC intersects the circumscribed circle at point N . From point L to the sides AB and AC are drawn the perpendiculars LK and LM respec- tively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKN M .3. Suppose x1 , x2 , . . . , xn are real numbers with x2 + x2 + · · · + x2 = 1. Prove 1 2 n that for any integer k > 1 there are integers ei not all 0 and with |ei | < k such that √ (k − 1) n |e1 x1 + e2 x2 + · · · + en xn | ≤ . kn − 1 Second Day (July 11)4. Does there exist a function f : N → N, such that f (f (n)) = n + 1987 for every natural number n?5. Prove that for every natural number n ≥ 3 it is possible to put n points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area.6.3.28.2 Longlisted Problems1. (AUS 1) Let x1 , x2 , . . . , xn be n integers. Let n = p + q, where p and q are positive integers. For i = 1, 2, . . . , n, put Si = xi + xi+1 + · · · + xi+p−1 and Ti = xi+p + xi+p+1 + · · · + xi+n−1 (it is assumed that xi+n = xi for all i). Next, let m(a, b) be the number of indices i for which Si leaves the remainder a and Ti leaves the remainder b on division by 3, where a, b ∈ {0, 1, 2}. Show that m(1, 2) and m(2, 1) leave the same remainder when divided by 3.
3.28 IMO 1987 2052. (AUS 2) Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A perfect shuffle of these cards changes the order to n + 1, 1, n + 2, 2, . . . , n − 1, 2n, n; i.e., the cards originally in the first n positions have been moved to the places 2, 4, . . . , 2n, while the remaining n cards, in their original order, fill the odd positions 1, 3, . . . , 2n − 1. Suppose we start with the cards in the above order 1, 2, . . . , 2n and then successively apply perfect shuffles. What conditions on the number n are necessary for the cards eventually to return to their original order? Justify your answer. Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.3. (AUS 3) A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop X to bus stop Y = X, then we shall say Y can be reached from X. We shall use the phrase Y comes after X when we wish to express that every bus stop from which the bus stop X can be reached is a bus stop from which the bus stop Y can be reached, and every bus stop that can be reached from Y can also be reached from X. A visitor to this town discovers that if X and Y are any two different bus stops, then the two sentences "Y can be reached from X" and "Y comes after X" have exactly the same meaning in this town. Let A and B be two bus stops. Show that of the following two statements, exactly one is true: (i) B can be reached from A; (ii) A can be reached from B.4. (AUS 4) Let a1 , a2 , a3 , b1 , b2 , b3 be positive real numbers. Prove that (a1 b2 + a2 b1 + a1 b3 + a3 b1 + a2 b3 + a3 b2 )2 ≥ 4(a1 a2 + a2 a3 + a3 a1 )(b1 b2 + b2 b3 + b3 b1 ) and show that the two sides of the inequality are equal if and only if a1 /b1 = a2 /b2 = a3 /b3 .5. (AUS 5) Let there be given three circles K1 , K2 , K3 with centers O1 , O2 , O3 respectively, which meet at a common point P . Also, let K1 ∩ K2 = {P, A}, K2 ∩ K3 = {P, B}, K3 ∩ K1 = {P, C}. Given an arbitrary point X on K1 , join X to A to meet K2 again in Y , and join X to C to meet K3 again in Z. (a) Show that the points Z, B, Y are collinear. (b) Show that the area of triangle XY Z is less than or equal to 4 times the area of triangle O1 O2 O3 .6. (AUS 6) (SL87-1).
3.28 IMO 1987 20716. (FRA 3) Let ABC be a triangle. For every point M belonging to segment BC we denote by B and c the orthogonal projections of M on the straight lines AC and BC. Find points M for which the length of segment B C is a minimum.17. (FRA 4) Consider the number α obtained by writing one after another the decimal representations of 1, 1987, 19872, . . . to the right the decimal point. Show that α is irrational.18. (FRA 5) (SL87-4).19. (FRG 1) (SL87-14).20. (FRG 2) (SL87-15).21. (FRG 3) (SL87-16).22. (GBR 1) (SL87-5).23. (GBR 2) A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance d from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. √ Prove that the area of the lampshade if d2 (2θ + 3), where cot θ = 3 . 2 θ24. (GBR 3) Prove that if the equation x4 + ax3 + bx + c = 0 has all its roots real, then ab ≤ 0.25. (GBR 4) Numbers d(n, m), with m, n integers, 0 ≤ m ≤ n, ae defined by d(n, 0) = d(n, n) = 0 for all n ≥ 0 and md(n, m) = md(n − 1, m) + (2n − m)d(n − 1, m − 1) for all 0 < m < n. Prove that all the d(n, m) are integers.26. (GBR 5) Prove that if x, y, z are real numbers such that x2 +y 2 +z 2 = 2, then x + y + z ≤ xyz + 2.27. (GBR 6) Find, with proof, the smallest real number C with the following property: For every infinite sequence {xi } of positive real numbers such that x1 + x2 + · · · + xn ≤ xn+1 for n = 1, 2, 3, . . . , we have √ √ √ √ x1 + x2 + · · · + xn ≤ c x1 + x2 + · · · + xn for n = 1, 2, 3, . . . .28. (GDR 1) In a chess tournament there are n ≥ 5 players, and they have 2 already played n + 2 games (each pair have played each other at most 4 once). (a) Prove that there are five players a, b, c, d, e for which the pairs ab, ac, bc, ad, ae, de have already played.
208 3 Problems n2 (b) Is the statement also valid for the 4 + 1 games played? Make the proof by induction over n.29. (GDR 2) (SL87-13).30. (GRE 1) Consider the regular 1987-gon A1 A2 . . . A1987 with center O. Show that the sum of vectors belonging to any proper subset of M = {OAj | j = 1, 2, . . . , 1987} is nonzero.31. (GRE 2) Construct a triangle ABC given its side a = BC, its circum- radius R (2R ≥ a), and the difference 1/k = 1/c − 1/b, where c = AB and b = AC.32. (GRE 3) Solve the equation 28x = 19y + 87z , where x, y, z are integers.33. (GRE 4) (SL87-6).34. (HUN 1) (SL87-8).35. (HUN 2) (SL87-9).36. (ICE 1) A game consists in pushing a flat stone along a sequence of squares S0 , S1 , S2 , . . . that are arranged in linear order. The stone is ini- tially placed on square S0 . When the stone stops on a square Sk it is pushed again in the same direction and so on until it reaches S1987 or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly n squares is 1/2n. Determine the probability that the stone will stop exactly on square S1987 .37. (ICE 2) Five distinct numbers are drawn successively and at random from the set {1, . . . , n}. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an 6 arithmetic progression is greater than (n−2)3 .38. (ICE 3) (SL87-10).39. (LUX 1) Let A be a set of polynomials with real coefficients and let them satisfy the following conditions: (i) if f ∈ A and deg f ≤ 1, then f (x) = x − 1; (ii) if f ∈ A and deg f ≥ 2, then either there exists g ∈ A such that f (x) = x2+deg g + xg(x) − 1 or there exist g, h ∈ A such that f (x) = x1+deg g g(x) + h(x); (iii) for every f, g ∈ A, both x2+deg f + xf (x) − 1 and x1+deg f f (x) + g(x) belong to A. Let Rn (f ) be the remainder of the Euclidean division of the polynomial f (x) by xn . Prove that for all f ∈ A and for all natural numbers n ≥ 1 we have Rn (f )(1) ≤ 0 and Rn (f )(1) = 0 ⇒ Rn (f ) ∈ A.40. (MON 1) The perpendicular line issued from the center of the circum- circle to the bisector of angle C in a triangle ABC divides the segment of
3.28 IMO 1987 209 the bisector inside ABC into two segments with ratio of lengths λ. Given b = AC and a = BC, find the length of side c.41. (MON 2) Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?42. (MON 3) Find the integer solutions of the equation √ √ 2 m = (2 + 2)n .43. (MON 4) Let 2n + 3 points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly n interior points is not less than 1 2n+3 . 3 244. (MOR 1) Let θ1 , θ2 , . . . , θn be real numbers such that sin θ1 + · · · + sin θn = 0. Prove that n2 | sin θ1 + 2 sin θ2 + · · · + n sin θn | ≤ . 445. (MOR 2) Let us consider a variable polygon with 2n sides (n ∈ N) in a fixed circle such that 2n − 1 of its sides pass through 2n − 1 fixed points lying on a straight line ∆. Prove that the last side also passes through a fixed point lying on ∆.46. (NET 1) (SL87-7).47. (NET 2) Through a point P within a triangle ABC the lines l, m, and n perpendicular respectively to AP, BP, CP are drawn. Prove that if l intersects the line BC in Q, m intersects AC in R, and n intersects AB in S, then the points Q, R, and S are collinear.48. (POL 1) (SL87-11).49. (POL 2) In the coordinate system in the plane we consider a convex polygon W and lines given by equations x = k, y = m, where k and m are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of W intersects a square if the boundary contains an interior point of the square. Prove that the boundary of W intersects at most 4 d unit squares, where d is the maximal distance of points belonging to W (i.e., the diameter of W ) and d is the least integer not less than d.50. (POL 3) Let P, Q, R be polynomials with real coefficients, satisfying P 4 +Q4 = R2 . Prove that there exist real numbers p, q, r and a polynomial S such that P = pS, Q = qS and R = rS 2 .
3.28 IMO 1987 213 2. (USA 3) At a party attended by n married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques C1 , C2 , . . . , Ck with the following prop- erty: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if n ≥ 4, then k ≥ 2n. 3. (FIN 3) Does there exist a second-degree polynomial p(x, y) in two variables such that every nonnegative integer n equals p(k, m) for one and only one ordered pair (k, m) of nonnegative integers? 4. (FRA 5) Let ABCDEF GH be a parallelepiped with AE BF CG DH. Prove the inequality AF + AH + AC ≤ AB + AD + AE + AG. In what cases does equality hold? 5. (GBR 1) Find, with proof, the point P in the interior of an acute-angled triangle ABC for which BL2 + CM 2 + AN 2 is a minimum, where L, M, N are the feet of the perpendiculars from P to BC, CA, AB respectively. 6. (GRE 4) Show that if a, b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then n−2 an bn cn 2 + + ≥ S n−1 , n ≥ 1. b+c c+a a+b 3 7. (NET 1) Given five real numbers u0 , u1 , u2 , u3 , u4 , prove that it is always possible to find five real numbers v0 , v1 , v2 , v3 , v4 that satisfy the following conditions: (i) ui − vi ∈ N. 0≤i<j≤4 (vi − vj ) < 4. 2 (ii) 8. (HUN 1) (a) Let (m, k) = 1. Prove that there exist integers a1 , a2 , . . . , am and b1 , b2 , . . . , bk such that each product ai bj (i = 1, 2, . . . , m; j = 1, 2, . . . , k) gives a different residue when divided by mk. (b) Let (m, k) > 1. Prove that for any integers a1 , a2 , . . . , am and b1 , b2 , . . . , bk there must be two products ai bj and as bt ((i, j) = (s, t)) that give the same residue when divided by mk. 9. (HUN 2) Does there exist a set M in usual Euclidean space such that for every plane λ the intersection M ∩ λ is finite and nonempty?10. (ICE 3) Let S1 and S2 be two spheres with distinct radii that touch externally. The spheres lie inside a cone C, and each sphere touches the cone in a full circle. Inside the cone there are n additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone C, both of the spheres S1 and S2 externally, as well as the two neighboring solid spheres. What are the possible values of n?
214 3 Problems11. (POL 1) Find the number of partitions of the set {1, 2, . . . , n} into three subsets A1 , A2 , A3 , some of which may be empty, such that the following conditions are satisfied: (i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. (ii) If A1 , A2 , A3 are all nonempty, then in exactly one of them the minimal number is even.12. (POL 5) Given a nonequilateral triangle ABC, the vertices listed coun- terclockwise, find the locus of the centroids of the equilateral triangles A B C (the vertices listed counterclockwise) for which the triples of points A, B , C ; A , B, C ; and A , B , C are collinear.13. (GDR 2)IMO5 Is it possible to put 1987 points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area?14. (FRG 1) How many words with n digits can be formed from the alphabet {0, 1, 2, 3, 4}, if neighboring digits must differ by exactly one?15. (FRG 2)IMO3 Suppose x1 , x2 , . . . , xn are real numbers with x2 + x2 + 1 2 · · · + x2 = 1. Prove that for any integer k > 1 there are integers ei not all n 0 and with |ei | < k such that √ (k − 1) n |e1 x1 + e2 x2 + · · · + en xn | ≤ . kn − 116. (FRG 3)IMO1 Let S be a set of n elements. We denote the number of all permutations of S that have exactly k fixed points by pn (k). Prove: n (a) k=0 kpn (k) = n!; n k=0 (k − 1) pn (k) = n!. 2 (b)17. (ROM 1) Prove that there exists a four-coloring of the set M = {1, 2, . . . , 1987} such that any arithmetic progression with 10 terms in the set M is not monochromatic. Alternative formulation. Let M = {1, 2, . . . , 1987}. Prove that there is a function f : M → {1, 2, 3, 4} that is not constant on every set of 10 terms from M that form an arithmetic progression.18. (ROM 4) For any integer r ≥ 1, determine the smallest integer h(r) ≥ 1 such that for any partition of the set {1, 2, . . . , h(r)} into r classes, there are integers a ≥ 0, 1 ≤ x ≤ y, such that a + x, a + y, a + x + y belong to the same class.19. (USS 2) Let α, β, γ be positive real numbers such that α + β + γ < π, α + β > γ, β + γ > α, γ + α > β. Prove that with the segments of lengths sin α, sin β, sin γ we can construct a triangle and that its area is not greater than 1 (sin 2α + sin 2β + sin 2γ). 8
3.28 IMO 1987 21520. (USS 3)IMO621. (USS 4)IMO2 The prolongation of the bisector AL (L ∈ BC) in the acute- angled triangle ABC intersects the circumscribed circle at point N . From point L to the sides AB and AC are drawn the perpendiculars LK and LM respectively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKN M .22. (VIE 3)IMO4 Does there exist a function f : N → N, such that f (f (n)) = n + 1987 for every natural number n?23. (YUG 2) Prove that for every natural number k (k ≥ 2) there exists an irrational number r such that for every natural number m, [rm ] ≡ −1 (mod k). Remark. An easier variant: Find r as a root of a polynomial of second degree with integer coefficients.
216 3 Problems3.29 The Twenty-Ninth IMOCanberra, Australia, July 9–21, 19883.29.1 Contest Problems First Day (July 15)1. Consider two concentric circles of radii R and r (R > r) with center O. Fix P on the small circle and consider the variable chord P A of the small circle. Points B and C lie on the large circle; B, P, C are collinear and BC is perpendicular to AP . (a) For which value(s) of ∠OP A is the sum BC 2 + CA2 + AB 2 extremal? (b) What are the possible positions of the midpoints U of BA and V of AC as OP A varies?2. Let n be an even positive integer. Let A1 , A2 , . . . , An+1 be sets having n elements each such that any two of them have exactly one element in common, while every element of their union belongs to at least two of the given sets. For which n can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly n/2 zeros?3. A function f defined on the positive integers (and taking positive integer values) is given by f (1) = 1, f (3) = 3, f (2n) = f (n), f (4n + 1) = 2f (2n + 1) − f (n), f (4n + 3) = 3f (2n + 1) − 2f (n), for all positive integers n. Determine with proof the number of positive integers less than or equal to 1988 for which f (n) = n. Second Day (July 16)4. Show that the solution set of the inequality 70 k 5 ≥ x−k 4 k=1 is the union of disjoint half-open intervals with the sum of lengths 1988.5. In a right-angled triangle ABC let AD be the altitude drawn to the hy- potenuse and let the straight line joining the incenters of the triangles ABD, ACD intersect the sides AB, AC at the points K, L respectively. If E and E1 denote the areas of the triangles ABC and AKL respectively, show that E1 ≥ 2. E6. Let a and b be two positive integers such that ab + 1 divides a2 + b2 . Show 2 2 that a +b is a perfect square. ab+1
3.29 IMO 1988 2173.29.2 Longlisted Problems 1. (BUL 1) (SL88-1). 2. (BUL 2) Let an = (n + 1)2 + n2 , n = 1, 2, . . . , where [x] denotes the integer part of x. Prove that (a) there are infinitely many positive integers m such that am+1 −am > 1; (b) there are infinitely many positive integers m such that am+1 −am = 1. 3. (BUL 3) (SL88-2). 4. (CAN 1) (SL88-3). 5. (CUB 1) Let k be a positive integer and Mk the set of all the integers that are between 2k 2 + k and 2k 2 + 3k, both included. Is it possible to partition Mk into two subsets A and B such that x2 = x2 ? x∈A x∈B 6. (CZS 1) (SL88-4). 7. (CZS 2) (SL88-5). 8. (CZS 3) (SL88-6). 9. (FRA 1) If a0 is a positive real number, consider the sequence {an } defined by a2 − 1 an+1 = n for n ≥ 0. n+1 Show that there exists a real number a > 0 such that: (i) for all real a0 ≥ a, the sequence {an } → +∞ (n → ∞); (ii) for all real a0 < a, the sequence {an } → 0.10. (FRA 2) (SL88-7).11. (FRA 3) (SL88-8).12. (FRA 4) Show that there do not exist more than 27 half-lines (or rays) emanating from the origin in 3-dimensional space such that the angle between each pair of rays is greater than of equal to π/4.13. (FRA 5) Let T be a triangle with inscribed circle C. A square with sides of length a is circumscribed about the same circle C. Show that the total length of the parts of the edges of the square interior to the triangle T is at least 2a.14. (FRG 1) (SL88-9).15. (FRG 2) Let 1 ≤ k < n. Consider all finite sequences of positive integers with sum n. Find T (n, k), the total number of terms of size k in all of these sequences.
218 3 Problems16. (FRG 3) Show that if n runs through all positive integers, f (n) = n+ n/3 + 1/2 runs through all positive integers skipping the terms of the sequence an = 3n2 − 2n.17. (FRG 4) Show that if n runs through all positive integers, f (n) = √ n + 3n + 1/2 runs through all positive integers skipping the terms of n2 +2n the sequence an = 3 .18. (GBR 1) (SL88-25).19. (GBR 2) (SL88-26).20. (GBR 3) It is proposed to partition the set of positive integers into two disjoint subsets A and B subject to the following conditions: (i) 1 is in A; (ii) no two distinct members of A have a sum of the form 2k + 2 (k = 0, 1, 2, . . .); and (iii) no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in a unique manner and determine the subsets to which 1987, 1988, and 1989 belong.21. (GBR 4) (SL88-27).22. (GBR 5) (SL88-28).23. (GDR 1) (SL88-10).24. (GDR 2) Let Zm,n be the set of all ordered pairs (i, j) with i ∈ {1, . . . , m} and j ∈ {1, . . . , n}. Also let am,n be the number of all those subsets of Zm,n that contain no two ordered pairs (i1 , j1 ), (i2 , j2 ) with |i1 − i2 | + |j1 − j2 | = 1. Show that for all positive integers m and k, m,2k ≤ am,2k−1 am,2k+1 . a225. (GDR 3) (SL88-11).26. (GRE 1) Let AB and CD be two perpendicular chords of a circle with center O and radius r, and let X, Y, Z, W denote in cyclical order the four parts into which the disk is thus divided. Find the maximum and minimum of the quantity A(YA(Z) ) , where A(U ) denotes the area of U . )+A(W27. (GRE 2) (SL88-12).28. (GRE 3) (SL88-13).29. (GRE 4) Find positive integers x1 , x2 , . . . , x29 , at least one of which is greater than 1988, such that x2 + x2 + · · · + x2 = 29x1 x2 . . . x29 . 1 2 2930. (HKG 1) Find the total number of different integers that the function
3.29 IMO 1988 219 5x f (x) = [x] + [2x] + + [3x] + [4x] 3 takes for 0 ≤ x ≤ 100.31. (HKG 2) The circle x2 + y 2 = r2 meets the coordinate axes at A = (r, 0), B = (−r, 0), C = (0, r), and D = (0, −r). Let P = (u, v) and Q = (−u, v) be two points on the circumference of the circle. Let N be the point of intersection of P Q and the y-axis, and let M be the foot of the perpendicular drawn from P to the x-axis. If r2 is odd, u = pm > q n = v, where p and q are prime numbers, and m and n are natural numbers, show that |AM | = 1, |BM | = 9, |DN | = 8, |P Q| = 8.32. (HKG 3) Assuming that the roots of x3 +px2 +qx+r = 0 are all real and positive, find a relation between p, q, and r that gives a necessary condition for the roots to be exactly the cosines of three angles of a triangle.33. (HKG 4) Find a necessary and sufficient condition on the natural num- ber n for the equation xn + (2 + x)n + (2 − x)n = 0 to have a real root.34. (HKG 5) Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.35. (HKG 6) In the triangle ABC, let D, E, and F be the midpoints of the three sides, X, Y , and Z the feet of the three altitudes, H the orthocenter, and P , Q, and R the midpoints of the line segments joining H to the three vertices. Show that the nine points D, E, F, P, Q, R, X, Y, Z lie on a circle.36. (HUN 1) (SL88-14).37. (HUN 2) Let n points be given on the surface of a sphere. Show that the surface can be divided into n congruent regions such that each of them contains exactly one of the given points.38. (HUN 3) In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any 3 of them there was a question that the three students answered differently. What is the maximal possible number of students tested?39. (ICE 1) (SL88-15).40. (ICE 2) A sequence of numbers an , n = 1, 2, . . ., is defined as follows: a1 = 1/2, and for each n ≥ 2, 2n − 3 an = an−1 . 2n n Prove that k=1 ak < 1 for all n ≥ 1.
220 3 Problems41. (INA 1) (a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the midpoint of side BC and points E and F are chosen on sides AC and AB respectively, and suppose that the lines EF and AM intersect at G. If AE = 2AF then find the ratio EG/GF . (b) Let E be a point external to a circle and suppose that two chords EAB and ECD meet at an angle of 40◦ . If AB = BC = CD, find the size of ∠ACD.42. (INA 2) (a) Four balls of radius 1 are mutually tangent, three resting an the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length s, is circumscribed around the balls. Find the value of s. (b) Suppose that ABCD and EF GH are opposite faces of a rectangu- lar solid, with ∠DHC = 45◦ and ∠F HB = 60◦ . Find the cosine of ∠BHD.43. (INA 3) (a) The polynomial x2k + 1 + (x + 1)2k is not divisible by x2 + x + 1. Find the value of k. (b) If p, q, and r are distinct roots of x3 − x2 + x − 2 = 0, find the value of p3 + q 3 + r3 . (c) If r is the remainder when each of the numbers 1059, 1417, and 2312 is divided by d, where d is an integer greater than one, find the value of d − r. (d) What is the smallest positive odd integer n such that the product of 21/7 , 23/7 , . . . , 2(2n+1)/7 is greater than 1000?44. (INA 4) (a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the polynomial g(x12 ) is divided by the polynomial g(x)? (b) If k is a positive integer and f is a function such that for every positive √ √ 2 12/y number x, f (x2 + 1) x = k, find the value of f 9+yy2 for every positive number y. (c) The function f satisfies the functional equation f (x) + f (y) = f (x + y) − xy − 1 for every pair x, y of real numbers. If f (1) = 1, find the number of integers n for which f (n) = n.45. (INA 5) (a) Consider a circle K with diameter AB, a circle L tangent to AB and to K, and a circle M tangent to circle K, circle L, and AB. Calculate the ratio of the area of circle K to the area of circle M . (b) In triangle ABC, AB = AC and CAB = 80◦ . If points D, E, and F lie on sides BC, AC, and AB, respectively, and CE = CD and BF = BD, find the measure of EDF .
222 3 Problems54. (KOR 2) (SL88-22).55. (KOR 3) Find all positive integers x such that the product of all digits of x is given by x2 − 10x − 22.56. (KOR 4) The Fibonacci sequence is defined by an+1 = an + an−1 (n ≥ 1), a0 = 0, a1 = a2 = 1. Find the greatest common divisor of the 1960th and 1988th terms of the Fibonacci sequence.57. (KOR 5) Let C be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners of C, keeping the new faces perpendicular to the diagonals of the cube and keeping the newly formed faces identical. If at the conclusion of this process the fourteen faces so formed have the same area, find the area of each face of the new solid.58. (KOR 6) For each pair of positive integers k and n, let Sk (n) be the base-k digit sum of n. Prove that there are at most two primes p less than 20,000 for which S31 (p) is a composite number.59. (LUX 1) (SL88-18).60. (MEX 1) (SL88-19).61. (MEX 2) Prove that the numbers A, B, and C are equal, where we define A as the number of ways that we can cover a 2 × n rectangle with 2 × 1 rectangles, B as the number of sequences of ones and twos that add up to n, and C as m 0 + m+1 2 + · · · + 2m 2m if n = 2m, m+1 1 + m+2 3 + · · · + 2m+1 2m+1 if n = 2m + 1.62. (MON 1) The positive integer n has the property that in any set of n integers chosen from the integers 1, 2, . . . , 1988, twenty-nine of them form an arithmetic progression. Prove that n > 1788.63. (MON 2) Let ABCD be a quadrilateral. Let A BCD be the reflection of ABCD in BC, while A B CD is the reflection of A BCD in CD and A B C D is the reflection of A B CD in D A . Show that if the lines AA and BB are parallel, then ABCD is a cyclic quadrilateral.64. (MON 3) Given n points A1 , A2 , . . . , An , no three collinear, show that the n-gon A1 A2 . . . An can be inscribed in a circle if and only if A1 A2 · A3 An · · · An−1 An + A2 A3 · A4 An · · · An−1 An · A1 An + · · · +An−1 An−2 · A1 An · · · An−3 An = A1 An−1 · A2 An · · · An−2 An .65. (MON 4) (SL88-20).
3.29 IMO 1988 22366. (MON 5) Suppose αi > 0, βi > 0 for 1 ≤ i ≤ n (n > 1) and that n n i=1 αi = i=1 βi = π. Prove that n n cos βi ≤ cot αi . i=1 sin αi i=167. (NET 1) Given a set of 1988 points in the plane, no three points of the set collinear, the points of a subset with 1788 points are colored blue, and the remaining 200 are colored red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100 red points.68. (NET 2) Let S be the set of all sequences {ai | 1 ≤ i ≤ 7, ai = 0 or 1}. The distance between two elements {ai } and {bi } of S is defined as 7 i=1 |ai − bi |. Let T be a subset of S in which any two elements have a distance apart greater than or equal to 3. Prove that T contains at most 16 elements. Give an example of such a subset with 16 elements.69. (POL 1) For a convex polygon P in the plane let P denote the convex polygon with vertices at the midpoints of the sides of P . Given an integer area(P ) n ≥ 3, determine sharp bounds for the ratio over all convex area(P ) n-gons P .70. (POL 2) In 3-dimensional space a point O is given and a finite set A of segments with the sum of the lengths equal to 1988. Prove that there exists a plane disjoint from A such that the distance from it to O does not exceed 574.71. (POL 3
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...
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This Book
basis for our contemporary digital computer systems. The book discusses direct theories of finite differences and integration, linear equations, variations of a constant, and equations of partial and mixed differences. Boole also includes exercises for daring students to ponder, and also supplies answers. Long a proponent of positioning logic firmly in the camp of mathematics rather than philosophy, Boole was instrumental in developing a notational system that allowed logical statements to be symbolically represented by algebraic equations. One of history's most insightful mathematicians, Boole is compelling reading for today's student of logic and Boolean
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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This field guide contains a quick look at the functions commonly encountered in single variable calculus, with exercises for each topic: linear, polynomial, power, rational, exponential, logarithmic, trigonometric, and...
ExploreLearning offers a catalog of modular, interactive simulations in math and science for teachers and students. They call these simulations Gizmos and they are meant to be fun, easy to use, and flexible enough to...
An introduction to magnetic resonance imagining, which is based on the principles of nuclear magnetic resonance (NMR), a spectroscopic technique used by scientists to obtain microscopic chemical and physical information...
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CBSE Board Syllabus for Math
CBSE maths syllabus 2013 in designed in a way to embellish the analytical and methodological skills, develop knowledge and mastery in various fields in Mathematics such as algebra, number system, trigonometry, statistics and geometry. The objective is to explicate reasoning and logical abilities, help the students in applying the knowledge they have gained with practical examples and to develop interest in subject with motivation and visualization.
CBSE math syllabus 2013 has been divided term wise according to CCE (Continuous and Comprehensive Evaluation) guidelines. There will be four Formative and two Summative assessments overall. The distribution of marks is as follows:
- 10 marks each for all Formative assessments (wherein two will be included in first term and two in second terms)
- 20 marks for first Summative assessment and 40 for second.
The course is structured to acquaint students with math and its applications in day to day life. According to guidelines for CBSE maths syllabus, it is ascertained that the subject is taught in a stimulating manner. The use of activities such as mathematical experiments and models, graphics, puzzles and games in encouraged. The syllabus aims at developing the algebraic abilities and making students familiar with problems of trigonometry, especially heights and distances. The students are made familiar and encouraged to experiment with numbers and geometric concepts. By laying attention on practical and theoretical topics, utilization of concepts learned is focused.
COURSE STRUCTURE
MATHEMATICS
CLASS 6
Number System
(i) Knowing our Numbers:
Consolidating the sense of numberness up to 5 digits, Size, estimation of numbers, identifying smaller, larger, etc. Place value (recapitulation and extension), connectives: use of symbols =, <, > and use of brackets, word problems on number operations involving larg...Read More
COURSE STRUCTURE
CLASS 8
MATHEMATICS
Number System
(i) Rational Numbers:
• Properties of rational numbers.(including identities). Usinggeneral form of expression to describe properties
• Consolidation of operations on rational numbers.
• Representation of rational numbers on the number line
• Between any two rational numbers there lies another rati...Read More
6. MATHEMATICS (Code No 041)
The Syllabus in the subject of Mathematics has undergone changes from time to time in
accordance with growth of the subject and emerging needs of the society. Senior Secondary stage is a
launching stage from where the students go either for higher academic education in Mathematics or for
professional courses like engineering, phy
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Student Handbook
Apply to Math 100 Positions
Math 100 Course Website
Welcome to the Math 100 course website!
This website is currently undergoing some construction, but there are links that you will find useful, whether you are a student currently taking Math 100 or Math 100AX, you are an advisor, or if you are just curious about the course.
Math 100 and Math 100AX are designed to bring the student up to speed on basic math skills, and to help the student be successful in their future college math classes. It is an online class; students from all over campus (and elsewhere) log in to a virtual classroom at the specified class time. Here, they are working closely with our instructional team to practice the skills that they either never fully learned, or have just forgotten.
On this website, you will find:
FAQs - a growing list of the most frequently asked questions about the course
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More About
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Overview
This book is a text for a one-semester course for upper-level undergraduates and beginning graduate students in engineering, science, and mathematics. Prerequisites are a first course in the theory of ODEs and a survey course in numerical analysis, in addition to specific programming experience, preferably in MATLAB, and knowledge of elementary matrix theory. Professionals will also find that this useful concise reference contains reviews of technical issues and realistic and detailed examples. The programs for the examples are supplied on the accompanying web site and can serve as templates for solving other problems. Each chapter begins with a discussion of the "facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed, but only those methods most widely used. The treatment of each method is brief and technical issues are minimized, but all the issues important in practice and for understaning the codes are discussed. The last part of each chapter is a tutorial that shows how to solve problems by means of small, but realistic, examples.
Editorial Reviews
From the Publisher
"The authors provide an excellent treatment of the fundamentals for solving ODEs using MATLAB. Their vast experience from research, the solution of real problems, and the teaching of this material is evident throughout the book." Mathematical Reviews
"...a very useful book to those who are looking to expand their MATLAB experience." Amanda Bligh, Product Design Engineer, Hasbor
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Calculus
It's a word that strikes fear into the most conscientious of students: calculus! But as Tim and Moby demonstrate in this BrainPOP movie, there's really nothing to be afraid of. You'll learn how calculus is used to measure things that are constantly changing, like irregular shapes and curvy lines. Rather than bog down in equations and jargon, Tim will explain the basic concepts behind calculus. He'll show you how calculus is really just a way of magnifying a complex system to break it up into billions of simple parts. You'll see how when you get close enough to a curved line, it actually straightens out! And you'll learn about the two main branches of calculus, differential and integral. Finally, Tim will go over some of the many ways in which calculus is used every day!
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Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Six hundred twenty three Linear Algebra textbooks are available for sale on ValoreBooks.com, twenty used from the cheapest price of $38.58, or buy new starting at $119.21
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Prealgebra - 4th edition
Summary: Prealgebra, 4/e, helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the rest of the text. This text's single greatest strength is the Aufmann Interactive Method, which enables students to work with the math concepts as they're being introduced. In addition, the text strongly emphasizes the AMATYC standards, with ...show morea special focus on real-sourced data.
New! AIM for Success Student Preface explains how students can achieve success using this text, and walks them through the use of specific supportive features. Prep Tests at the beginning of each chapter focus on the particular prerequisite skills that will be used in the upcoming chapter. Go Figure problems are puzzles designed to engage students in problem solving.
Aufmann Interactive Method provides students with an opportunity to try a skill as it is presented. Each section is divided into objectives, and every objective contains one or more sets of matched-pair examples. The first example in each set is worked out; the second example, called You Try It, requires students to actively practice concepts as they are presented in the text. Complete worked-out solutions to these examples are in the appendix, providing immediate feedback and reinforcement.
Engaging chapter-opening photos illustrate and reference a specific mathematical application from the chapter. Short-answer Critical Thinking exercises and writing exercises at the end of each section reinforce understanding by encouraging students to think about or review an important concept presented in the lesson.
All of the Exercise Sets emphasize skill building, skill maintenance, and applications.
End-of-chapter support includes Focus on Problem Solving, which introduces students to various problem-solving strategies, and Projects & Group Activities, which include such varied applications as music, the stock market, earned run averages, buying a car, and nutrition.
Throughout the text, data problems have been updated to reflect current data and trends. These application problems demonstrate for students the variety of problems that require mathematical analysis. Instructors can use many of these problems to lead interesting class discussions.
Take Note margin notes alert student to a point requiring special attention or amplify the concept under discussion. Point of Interest margin notes contain interesting sidelights about mathematics, its history, or its application. Calculator Notes provide suggestions for using a calculator in certain situations. Calculator icons in both the Student Edition and Instructor's Annotated Edition indicate exercises that require a calculator.
For ease of navigation and flexibility in choosing applications, an Index of Application appears on the inside front cover, and a chapter-by-chapter index of application problems can be found in the Table of Contents.
Chapter Review Exercises at the end of each chapter help students integrate all of the topics presented in the chapter. A two-page Chapter Test provides students with the opportunity to test their understanding of the material and to diagnose their weaknesses. Cumulative Review Exercises help students maintain skills learned in previous chapters.
Revised! Chapter Summaries have been rewritten to be a more useful guide to students as they review for a test. These sections review Key Words and highlight the Essential Rules and Procedures that were introduced in the chapter, accompanied by examples.
Section 3.1 Least Common Multiple and Greatest Common Factor Section 3.2 Introduction to Fractions Section 3.3 Addition and Subtraction of Fractions Section 3.4 Multiplication and Division of Fractions Section 3.5 Solving Equations with Fractions Section 3.6 Exponents, Complex Fractions, and The Order of Operations Agreement Focus on Problem Solving: Common Knowledge Projects and Group Activities: Music/Construction/Fractions of Diagrams/Puzzle from the Middle Ages/Using Patterns in Experiment
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Problem Solvers are an exceptional series of books that are thorough, unusually well-organized, and structured in such a way that they can be used with any text. No other series of study and solution guides has come close to the Problem Solvers in usefulness, quality, and effectiveness. Educators consider the Problem Solvers the most effective series of study aids on the market. Students regard them as most helpful for their school work and studies. With these books, students do not merely memorize the subject matter, they really get to understand it. Each Problem Solver is over 1,000 pages, yet each saves hours of time in studying and finding solutions to problems. These solutions are worked out in step-by-step detail, thoroughly and clearly. Each book is fully indexed for locating specific problems rapidly. For linear algebra courses, as well as for courses in computers, physics, engineering, and sciences which use linear algebra. Concentrations on solutions to applied problems in economics, mechanics, electricity, chemistry, geometry, business, probability, graph theory, and linear programming.
Most Helpful Customer Reviews
It is a good book filled with alot of information. I needed the book for a course I am taking in Math Anaylsis to help me prepare for the test I have to take. Math Analsis includes both algebra and calculus. It gives you the problem on top of the page and then gives you step by step instructions which is very good for slow math learners lke myself.
If you are one of a handful of people who are attempting to pass the CSET Single Subject Mathematics Test, listen up. First, I have passed the CSET Math Section II exam. I am in the process of trying to pass the CSET Math Section I exam. While I still need to take it again, I now know the secret because the only difference between my first try on section I and second try was an 8 page section of the Linear Algebra Problem Solver book. My score went up 20 points and I KNOW the book is the reason. Now, I am trying to track down the problems from the REA Pre-Calculus Problem Solver book having to do with growth rates and logarithms..this will undoubtedly put me 'over the top'. These books work and I am now 100% sure that the exam writers are a subset of authors who helped put the problem solvers together. Here is how to use them if you did not pass the exam on the first try: 1) Look through them and pinpoint the questions that look familar to you from your first (or second) experiences taking the exam. 2) Know the 3-4 questions from these books (cold)which are almost exactly the same as the exam questions. In my case, this comes from a mere 8 page section found on ppg. 481-488 of the REA Linear Algebra Problem Solver text. 3) Supplement this with information from [...] and download all 40 pages or so of the math listed there. 4) Finally, know the example section from the CSET site COLD, particularly the essay section. Take the Field theory answer and turn it into a matrix so that you can formulate an answer quickly on the actual exam. I was able to pull off section II of the CSET and had a vague idea of what I was doing at the time. Now that I have had time to review my own process and recall the questions, I am sure there is a method to my madness.Read more ›
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More About
This Textbook
Overview
An understanding of discrete mathematics is essential for students of computer science wishing to improve their programming competence.
Fundamentals of Discrete Math for Computer Science provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions.
Topics and features:
Highly accessible and easy to read, introducing concepts in discrete mathematics without requiring a university-level background in mathematicsIdeally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendationsDescribes mathematical processes in an algorithmic manner, often including a walk-through demonstrating how the algorithm performs the desired task as expectedContains examples and exercises throughout the text, and highlights the most important concepts in each sectionSelects examples that demonstrate a practical use for the concept in questionThis easy-to-understand and fun-to-read textbook is ideal for an introductory discrete mathematics course for computer science students at the beginning of their studies. The book assumes no prior mathematical knowledge, and discusses concepts in programming as needed, allowing it to be used in a mathematics course taken concurrently with a student's first programming course.
Editorial Reviews
From the Publisher
This book is specifically aimed at CS students. The authors include the same discrete math topics that other books have, but, in contrast to most existing books, they introduce each topic with a clear (and entertaining) CS motivation…
The book covers the usual discrete math topics …in a very entertaining way…
Each section is well written, with a highlighted subsection on the most important ideas and plenty of exercises. I highly recommend this book to everyone. It can be used in two different ways. The easiest way is to teach only the topics that are usually taught in discrete math classes (and ignore the other parts of the book). Alternatively, you could cover the whole book and, if needed, rearrange the other classes to avoid duplication. No matter how you use this book, its highly entertaining presentation of the material will undoubtedly make the class a success.
"Jenkyns (Brock Univ., Canada) and Stephenson (Univ. of Calgary, Canada) have written an introductory textbook on discrete mathematics for computer science majors. The volume's ten chapters cover the standard topics taught in such courses at the freshman or sophomore level … . In comparison with other introductory discrete mathematics textbooks, this work has a very strong emphasis on algorithms, proofs of algorithmic correctness, and the analysis of worst-case and average-case complexity. … Summing Up: Recommended. Lower-division undergraduates." (B. Borchers, Choice, Vol. 50 (9), May, 2013)
Meet the Author
Dr. Tom Jenkyns is an Associate Professor in the Department of Mathematics and the Department of Computer Science at Brock University, Canada.
Dr. Ben Stephenson is an Instructor in the Department of Computer Science at the University of Calgary
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Twelfth graders investigate the capabilities of the TI-89. In this calculus lesson, 12th graders explore the parametric equation for a circle, for arc length of curves, and for trajectories. Students investigate the symbolic and graphical representation of vectors. Students use polar functions of investigate the area bounded by a curve. Students investigate a 3D graphing application.
In this derivatives worksheet, students sketch the graph of a function by determining the local maxima, minima and inflection points. They graph a second function when given the interval. Students solve two word problems involving maximum area.
In this calculus worksheet, students solve ten multiple-choice questions and eleven free-response questions covering limits, derivatives, and integrals. Students should have covered trig limits prior to this worksheet.
In this calculus worksheet, students observe graphs and identify the limits of the functions listed in the graph. They determine the definite integrals and derivatives. Students use the trapezoid rule to estimate distance. This five-page worksheet contains 14 problems.
In this rate equation worksheet, students identify an inflection value for given functions. They determine an expression that stultifies a given rate equation. This three-page worksheet contains definitions, explanations and examples. It provides three multi-step practice problems, with graphs.
In this function of two variables worksheet, students explore the relationship between the derivatives of a function and the shape of the graph. They determine the maximum and minimum value using optimization. Students find the equation of a vertical slice of an object and sketch a picture of the surface graph. This four-page worksheet contains eight multi-step functions.
In this harmonic function learning exercise, students explore harmonic functions and construct a harmonic conjugate. This three-page learning exercise contains three problems, with explanations and examples.
In this function worksheet, students sketch the graph of a nonzero function, estimate the value of an infinite sum, and determine the derivative and slope of functions. This one-page worksheet contains approximately 6 multi-step problems.
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled �Proof of Derivatives of Ln(x) and e^x,� has a clearer explanation of this proof.
Students research the functions of a cell through using a variety of activities. They are focused on the end result of creating a model of the cell that is labeled with the parts of a cell and how they function.
Students solve equations with combination and functions. For this algebra lesson, students identify and create formulas to a graph and word problem. They perform operation when solving combination of a function.
In this accumulation functions worksheet, students describe the graph of an accumulation function for a constant function. Students describe the link between the slope of the graph of the accumulation function and the value of the function.
In this function activity, students use various methods to solve functions. They explore the logarithm function, the derivative of an exponential function, and compose a function with a linear equation. This four-page activity contains explanations, examples, and four problems.
In this complex function worksheet, students examine the properties of integrals of complex functions of a real variable, smooth arcs, and the Jordan Curve Theorem. This two-page worksheet contains approximately seven equations.
Students find the slope and equation of a line. In this algebra instructional activity, students are able to graph lines given a table of values and are able to derive the slope and intercept given an equation. They evaluate coordinate pairs and identify functions.
Students investigate the properties of inverse functions. In this trigonometry lesson, students write trigonometric equations for given functions. They calculate the inverse using properties of sine, cosine and tangent.
Young scholars identify the six trig values using ratios of a right triangle. In this trigonometry lesson, students graph inverse functions and perform operation using inverse trig values. They identify different properties of trig graphs.
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Discrete mathematics is a blend of many different elements of logic, combinatorics and graph theory. I hold a Master's in Math Education and have coached many students through various Discrete math courses. Let me help you reduce your math anxiety
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9780130400 algebra-based Introductory Statistics courses. Offering the most accessible approach to statistics, with a strong visual/graphical emphasis, this text offers a vast number of examples on the premise that students learn best by "doing". The fourth edition features many updates and revisions that place increased emphasis on interpretation of results and critical thinking in addition to calculations. This emphasis on "statistical literacy" is reflective of the GAISE recommendations. Datasets and other resources (where applicable) for this book are available
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Clifton Park 2002 Softcover Second Printing. 360 pages. Softcover. Brand new book. MATHEMATICS. Written for students, teachers, tutors, and parents, as well as for scientists ...and engineers. It offers an application-oriented focus and includes: A revieew of basic geometry and definitions pertaining to triangles, the trigonometric functions, circular functions, and graphs; The periodic nature of trigonometric functions, inverse trigonometric functions, trigonometric identities, trigonometric equations and inequalities, vectors, polar coordinates, and complex numbers; Relationships between trigonometric functions and exponential functions, hyperbolic functions, and series expansions, as well as spherical trigonometry. Includes an Index. "One of the most extensive treatments of fundamental trigonometry skills available-user-friendly and definitely a must for learning trigonometry basics or for those who want a comprehensive refresher."-Channing Robertson, Ruth G. and William Bowes, Professors in the School ofRead moreShow Less
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Overview
Master Math: Trigonometry is written for students, teachers, tutors, and parents, as well as for scientists and engineers who need to look up principles, definitions, explanations of concepts, and examples pertaining to the field of trigonometry. Trigonometry is a visual and application-oriented field of mathematics that was developed by early astronomers and scientists to understand, model, measure, and navigate the physical world around them.
Related Subjects
Meet the Author
Debra Anne Ross has a double BA in Chemistry and Biology from the University of California, Santa Cruz, and and MS in Chemical Engineering from Stanford University. Debra's career encompasses biology, chemistry, biochemistry, engineering, biosensors, pharmaceutical drug discovery, and intellectual property. She is the author of the popular Master Math books, The 3:00 PM Secret: Live Slim and Strong Live Your Dreams, The 3:00 PM Secret: Ten Day Dream Diet (2009), and Arrows Through Time: A Time Travel Tale of Adventure, Courage, and Faith Algebra Book Available!
This is the best book on learning basic algebra. It is thorough yet concise. The information is presented very clearly. The author has obviously tried to explain the concepts so that they `make sense¿ to students - and their parents. I use the book to explain algebra to my students. Like the other Master Math books by Ross, the topics flow logically and build in difficulty. What a breath of fresh air after the often confusing text books students are given in school. This book is helpful for students struggling with algebra and the parents who are tutoring them. This book is also extremely useful for older students who did not adequately learn algebra, yet find they need to know it later. Topics can easily be looked up and reviewed or learned. I highy recommend this book!
5 out of 7 people found this review helpful.
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Log-IC
Posted October 10, 2009
ALG 101 Launchpad
Splendid clarity and progression.
A cherry on top of this fine sundae,
maybe a single page of exponential,
radical,and complex properties for
a quick reference,( but that's nitpicking).
3 out of 5 people found this review helpful.
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Anonymous
Posted November 20, 2007
Outstanding Trigonometry Presentation!
This is the best book out there on learning trigonometry. I especially appreciate the visually-oriented focus. Each concept is described in all its forms, such as sine. Do you know each of the different ways sine can be described? Like the other Master Math books by Ross, the topics flow logically and are in context with what precedes and follows. It is thorough yet concise, and packed full of everything you, as tutor, or your kids need to know. The real world and fun applications are wonderful! The information is explained clearly and in a way that makes sense, so that a given concept is explained in such a way you understand what is being discussed rather than just memorizing formulas. What a breath of fresh air after the often confusing text books I was and my children are given in school. I really feel I can explain trigonometry to young people using this book! if I were going back to school, and taking math or science, this book would be in my backpack.
3 out of 5 people found this review helpful.
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Anonymous
Posted November 21, 2012
page 14 e reader
i am not sure if i am missing something but on page 14 the book says x times $1.00 per glass will equal 20. then shows this equation
X + ($1.00 per glass)=$20.00
1 out of 1 people found this review helpful.
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Anonymous
Posted February 12, 2012
the internet is more helpful
This book is terrible. There are many books that are better. I found websites more helpful.
1 out of 3 people found this review helpful.
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Matrix Multiplier
This resource is a tool for manipulating 2 matrices: any algebraic formula or solving a system of linear equations by matrix techniques. A reasonable calculator for 2 matrices, not just matrix multiplication. It is a little hard to navigate. It does all the computations for the student, so they don't have to learn how to do it.
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Multivariable Calculus
The Larson Calculus program has a long history of innovation in the calculus market. It has been widely praised by a generation of students and ...Show synopsisThe Larson Calculus
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This is a dynamic math website where students enter problems and where the site's math engine solves the problem. Students in most cases are given a step-by-step process for solving the problem that student enters immediately
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Editorial Reviews
Children's Literature
- Michael Chabin
I found this book impossible to read. Part of the reason is the prose, as exemplified by this passage: "Some multistepped problems require students to actually compute to find the answer, whereas other multistepped problems focus on notation and require the students to set up the series of equations needed or to find a single equation that combines all the steps together." That is English. I can take it apart, but if I do, it is a lot less important than it sounds and suggests an ominous absence of careful editing. Part of the reason for this lack of oversight is the book's purpose. It seems to have been designed to help teachers prepare their students for standardized tests. For those of us who think elementary teachers should focus on helping their students prepare for algebra, that is disturbing. Part of the reason is the author's notion of math itself as a collection of ever more complex and arbitrary algorithms one memorizes with the help of diagrams that are (to a child, anyway) as complex as the notion they are supposed to illustrate. That mistake, while common among educators, is one no mathematician would make. There is no better demonstration of the split between the practice and teaching of mathematics that has served American children so badly. Reviewer: Michael Chabin
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DVD Features:
Run Time: 35 minutes
Released: July 15, 2009
Originally Released: 2009
Label: Tmw Media Group
Encoding: Region [unknown]
Product Description:
Word problems lose some of their mystery with this helpful series geared toward algebra students. This step-by-step instructional focuses on number relations and demonstrates how to glean pertinent information from word problems to form the correct equations.
Plot Keywords:
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I have used this book both in class as part of the teaching and also to revise from at home in the run-up to the exam. Each topic is clearly labelled and everything is structured in order. Key points/facts are in bold to be easily identified by the reader. One of the best features of the book is that each topic contains WORKED EXAMPLES that show us the method used to obtain an answer. This is much more beneficial than just providing an answer. At the end of each topic there are exercises to test whether the knowledge has sunk in. The Review Exercises are especially helpful as they provide a huge selection of PAST EXAMINATION QUESTIONS to try. This, I think, is very good preparation for the real exam. All questions have answers at the back (although be wary there are a few mistakes) and also a P1 Examination Style Paper to try. For complete high quality learning materials you should also purchase REVISE FOR P1 MATHEMATICS as this contains even more worked examples and questions to try. Good luck with your studies and I hope the Heinemann suite of books increase your enjoyment for learning Mathematics. It certainly has mine!
This book really is excellent. The wording is clear and not excessive, the examples are useful learning aids and the exercise questions are well written, to gradually develop the expertise of the reader. The separate chapters cover groups of topics well, and separate the work into easily manageable sections, which are directly relevant to the syllabus of the Further Mathematics A Level. My only qualm is that some (only a few) of the answers to the many, many exercise questions are incorrect, which may have you scratching your head for a while in puzzlement. However, despite this flaw, this text really is superb, along with the others in this series. They got me an A! (With a little help from my teachers, of course.)
I am astonished by the numerous printing mistakes in the books. Just in Review exercise One, there are roughly 20 mistakes out of 117 questions, either in the questions or in the answers at the back. There are also serious errors in the the formulae (eg the section on compound angle in Book2). Beware if you rely too much on it. I wonder how many people like me who find out all the mistakes. If you can, I can almost assure you an A grade.
Mannall and Kenwood really ought to be ashamed of themselves for this effort. This book is of virtually no value to a student attempting to study P2 on their own. The only saving grace for this text book is that it contains a lot of practice questions. The most glaring problem is that anyones attempt to answer around 25% of those questions will not find any help / method or assistance in the book to do so. Sadly, the only 'clear route to a-level success' as suggested on the top of the book is to find someone who can help! This book will help identify what you do not understand without sufficient guidence to deal with problems. The sections labelled 'review questions' certainly do not 'review' the material that is covered in the method material. My adivce, steer clear - this is woefully inadequate.
This book isn't brilliant, it's not even good. It is of some help when accompanying lessons in pure maths, but in terms of learning from a book, look elsewhere, this one won't teach you anything. The examples are of little or no help with the exercises, and explanations, as with the Decision 1 book, are incomprehensible. Something that makes this book stand out from the others in a series plagued by mistakes is the sheer volume of errors, far more than the other books, there are errors in formulae, summaries, questions and exercises- not what the exam board would expect from a good maths student, and certainly not what you'd expect from the "experts".
Once I started to read the book, it was so direct to my Maths A-level for my particular syllabus. The practise questions,are relavent, but some topics such as the first one, don't have enough questions. comments by examiners who mark for this examining board, are very valuble. It may seem it bit blinkered to the world of Maths, but it can help most students to boost their grades. Unless of course you're already a top grade student! The material in this book usually ends up being tested for entrance exams into some Oxford unversities for subjects such as Physics, Maths, or Engineering. So it's worthwhile making sure that you know all the stuff, If you are required to sit a Maths test on the common core material, to get a place at college or university.
I'd reccomend this book to anyone considering doing AS Mathematics, although the specifications are changing slightly next year. I just sat my P1 exam a few hours ago and am relatively sure of an A/B grade. I also used the M1 book for my mechanics revision and found that exam easy too. Mathematics is a difficult A level to cover and ensure you understand in just 10 months worth of classes, this book makes it possible to work at home without any external aid. Much reccommended. I was going to drop maths at the start of the year but this book and some hard work has made P1/M1/S1 easy for me.
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Setting the standard in middle school mathematicsMathematics: Applications and Concepts is a three-course middle school series intended to bridge the gap from elementary mathematics to Algebra 1. The program is designed to motivate your students, enable them to see the usefulness of mathematics in the world around them, enhance their fluency in the language of mathematics, and prepare them for success in algebra and geometry.
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2014b.00403}
\itemau{Walkowiak, Temple A.}
\itemti{Elementary and middle school students' analyses of pictorial growth patterns.}
\itemso{J. Math. Behav. 33, 56-71 (2014).}
\itemab
Summary: Research has suggested the importance of incorporating algebraic thinking early and throughout the K-12 mathematics curriculum. One approach to help children develop algebraic reasoning is through the examination of pictorial growth patterns, which serve as a context for exploring generalization. The purpose of this study was to compare how elementary and middle school students analyze pictorial patterns, with a focus on whether students used figural or numerical reasoning. Task-based interviews were conducted with a second grader, fifth grader, and eighth grader in which they were asked to describe, extend, and generalize two pictorial growth patterns. Using a phenomenographic approach, analyses showed younger students used figural reasoning more than older students, but all students did not exclusively use figural or numerical reasoning. The students' generalizations included informal notation, descriptive words, and formal notation. The findings suggest that pictorial growth patterns are a promising tool for young students' development of algebraic thinking.
\itemrv{~}
\itemcc{F32 F33 H23 C32 C33}
\itemut{algebra; algebraic reasoning; patterns; geometric patterns; pictorial growth patterns; elementary mathematics; middle school mathematics; student thinking; phenomenography}
\itemli{doi:10.1016/j.jmathb.2013.09.004}
\end
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Beginning Algebra clear, accessible treatment of mathematics features a building-block approach toward problem solving, realistic and diverse applications, and chapter organizer to help users focus their study and become effective and confident problem solvers. The Putting Your Skills to Work and new chapter-end feature, Math in the Media, present readers with opportunities to utilize critical thinking skills, analyze and interpret data, and problem solve using applied situations encountered in daily life. Earlier coverage of the Order of Arithmetic Operations--now section 1.5 so that operations is now covered together before Introduction to Algebra. The discussion of solving linear equations in Chapter 2 now includes coverage of equations with no solution and equations with infinitely many solutions. Section 4.3 now offers a more thorough introduction to polynomials, with the addition of new terminology at the beginning of the section and a new lesson on evaluating polynomials at the end. Revised Ch. 7 on Graphing and Functions includes new coverage of the rectangular coordinate system and slope. The coverage of the rectangular coordinate system in Chapter 7 has been improved for greater clarity.
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