pe-nlp/ORZ-MATH-57k-Filter-difficulty · Datasets at Hugging Face

problem stringlengths 18 4.46k	answer stringlengths 1 942	pass_at_n float64 0.08 0.92
The three-digit number ends with the digit 2. If this digit is moved to the beginning of the number, the resulting number will be 18 greater than the original number. Find this number.	202	0.875
At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?	49	0.5
One ball is taken at random from a bag containing $(b-4)$ white balls and $(b+46)$ red balls. If $\frac{c}{6}$ is the probability that the ball is white, find $c$.	2	0.625
In a certain book, there were 100 statements written as follows: 1) "In this book, there is exactly one false statement." 2) "In this book, there are exactly two false statements." ... 3) "In this book, there are exactly one hundred false statements." Which of these statements is true?	99	0.875
A mouse is gnawing on a cube of cheese with an edge of 3, divided into 27 unit cubes. When the mouse eats a cube, it moves to another cube that shares a face with the previous one. Can the mouse eat the entire cube except for the central unit cube?	\text{No}	0.75
The side of an equilateral triangle inscribed in a circle is $a$. Calculate the area of the square inscribed in the same circle.	\frac{2 a^2}{3}	0.875
The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases.	150	0.875
A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different.	16	0.375
If \( 16^{\sin^2 x} + 16^{\cos^2 x} = 10 \), then \( \cos 4x = \) _______.	-\frac{1}{2}	0.75
In a single round-robin table tennis tournament, a) 10 players, b) 11 players competed. What is the minimum number of matches the champion must have won, given that there was no tie for first place?	6	0.625
There exists \( x_{0} < 0 \) such that \( x^{2} + \|x - a\| - 2 < 0 \) (where \( a \in \mathbb{Z} \)) is always true. Find the sum of all values of \( a \) that satisfy this condition.	-2	0.75
As shown in the figure, point $D$ is an interior point of $\triangle ABC$. $BD$ intersects $AC$ at $E$, and $CD$ intersects $AB$ at $F$, with $AF = BF = CD$ and $CE = DE$. Find the measure of $\angle BFC$.	120^\circ	0.625
Calculate the volume of a tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height from vertex \( A_{4} \) to the face \( A_{1}A_{2}A_{3} \). \( A_{1}(3, 10, -1) \) \( A_{2}(-2, 3, -5) \) \( A_{3}(-6, 0, -3) \) \( A_{4}(1, -1, 2) \)	7	0.375
Let \( P \) be a polynomial of degree 4 such that \( P(0) = P(1) = 1 \), \( P(2) = 4 \), \( P(3) = 9 \), and \( P(4) = 16 \). Calculate \( P(-2) \).	19	0.875
For \(x, y, z \geq 1\), find the minimum value of the expression $$ A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x} $$	1	0.875
A town has \( c \) celebrities and \( m \) weekly magazines. One week each celebrity was mentioned in an odd number of magazines, and each magazine mentioned an odd number of celebrities. Show that \( m \) and \( c \) are both even or both odd. How many ways are there for each magazine to mention an odd number of celebrities so that each celebrity is mentioned an odd number of times?	2^{(m-1)(c-1)}	0.625
Find the rank of the \( (2n+1) \times (2n+1) \) skew-symmetric matrix with entries given by: - \( a_{ij} = 1 \) for \( i - j = -2n, -(2n-1), \ldots, -(n+1) \) - \( a_{ij} = -1 \) for \( i - j = -n, -(n-1), \ldots, -1 \) - \( a_{ij} = 1 \) for \( i - j = 1, 2, \ldots, n \) - \( a_{ij} = -1 \) for \( i - j = n+1, n+2, \ldots, 2n+1 \) In other words, the main diagonal is zeros, the diagonals immediately below the main diagonal (up to \( n \) diagonals) are \( 1s \), the diagonals below those are \( -1s \), the diagonals immediately above the main diagonal (up to \( n \) diagonals) are \( -1s \), and the diagonals above those are \( 1s \).	2n	0.75
Is it possible to place 8 different numbers, none of which are divisible by 13, from the range of 1 to 245 at the vertices of a cube such that the numbers in adjacent vertices have a common divisor greater than 1, while the numbers in non-adjacent vertices do not share any common divisor greater than 1?	\text{No}	0.875
Let \( x \) be a non-zero real number such that \[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \]. Find the product of all possible values of \( x \).	-5	0.125
The manager of a store checked the sales price in 2006 of a television from the brand VejoTudo. He found a partially faded invoice that read: "lot of 72 VejoTudo TVs sold for R$ \ldots 679 \ldots reais," where the digits of the unit and the ten thousand place were illegible. What was the sales price in 2006 of each of those televisions?	511	0.625
\( N \) is a 5-digit number formed by 5 different non-zero digits, and \( N \) is equal to the sum of all 3-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers \( N \).	35964	0.75
Suppose that \( x - y = 1 \). Find the value of \( x^4 - xy^3 - x^3 y - 3x^2 y + 3xy^2 + y^4 \).	1	0.875
Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient?	231	0.125
Given a sequence of positive integers $\left\{a_{n}\right\}$ defined by $a_{0}=m$ and $a_{n+1}=a_{n}^{5}+487$ for $n \in \mathbf{N}$, find the value of $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares.	9	0.75
Find the smallest positive integer that can be expressed as the sum of 9 consecutive integers, 10 consecutive integers, and 11 consecutive integers.	495	0.625
Each cell in a \(5 \times 5\) grid contains a natural number written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of any three numbers inside any \(1 \times 3\) rectangle is 23. What is the value of the central number in the grid?	16	0.25
Masha was given a box with differently colored beads (each bead having a unique color, with a total of $\mathrm{n}$ beads in the box). Masha chose seven beads for her dress and decided to try out all possible combinations of these beads on the dress (thus, Masha considers sewing one, two, three, four, five, six, or seven beads, with the order of the beads not mattering to her). Then she counted the total number of combinations and was very surprised that the number turned out to be odd. 1) What number did Masha get? 2) Is it true that when selecting from a set with an even number of beads, Masha could get an even number of combinations? 3) Is it true that if the order of the beads sewn onto the dress mattered to Masha, she could have ended up with either an even or odd number of combinations?	\text{Yes}	0.125
Let \( a \) be a strictly positive integer. Suppose that \( 4(a^n + 1) \) is the cube of an integer for every positive integer \( n \). Find \( a \).	1	0.75
January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year?	52	0.75
Given a right triangle \( \triangle ABC \) with vertices \( A, B, \) and \( C \) on the parabola \( y = x^2 \), and the hypotenuse \( AB \) parallel to the x-axis, find the length of the altitude from \( C \) to the hypotenuse \( AB \).	1	0.875
Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.	10	0.375
What is the smallest number of participants that could have been in the school drama club if the fifth graders made up more than \(22\%\) but less than \(27\%\) of the club, the sixth graders made up more than \(25\%\) but less than \(35\%\), and the seventh graders made up more than \(35\%\) but less than \(45\%\) (there were no participants from other grades)?	9	0.25
Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.	272	0.375
If the complex number \( z \) satisfies \[ \|z-1\| + \|z-3-2i\| = 2\sqrt{2}, \] then the minimum value of \( \|z\| \) is ______ .	1	0.875
Find the equation of the normal line to the given curve at the point with the abscissa $x_{0}$. $y=\sqrt{x}-3 \sqrt[3]{x}, x_{0}=64$	x = 64	0.875
Given a cube with a side length of 4, is it possible to entirely cover three of its faces, which share a common vertex, using 16 paper rectangular strips each measuring \(1 \times 3\)?	\text{No}	0.625
Compute the limit of the numerical sequence: \[ \lim _{n \rightarrow \infty} \frac{\sqrt{n+1}-\sqrt[3]{n^{3}+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^{5}+1}} \]	1	0.75
Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers?	1665	0.125
Let \(a\) and \(b\) be coprime integers. Show that \(a + b\) and \(ab\) are coprime.	1	0.625
What number must be added to both terms of a fraction to obtain the reciprocal of that same fraction?	- (a + b)	0.875
Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards. What is the smallest value of \( n \) for which Zoé can ensure that she can play her two turns?	14	0.25
Two cars, Car A and Car B, simultaneously depart from point $A$ to point $B$. At the start, Car A’s speed is 2.5 km/h faster than Car B’s speed. After 10 minutes, Car A slows down; another 5 minutes later, Car B also slows down, and at this time, Car B is 0.5 km/h slower than Car A. After another 25 minutes, both cars arrive at point $B$ at the same time. By how many kilometers per hour did Car A's speed decrease?	10 \text{ km/h}	0.625
Place a known mathematical symbol between the numbers 4 and 5 such that the result is a number greater than 4 but less than 5.	4.5	0.625
The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM?	30 \text{ minutes}	0.75
Given a sequence \(\{a_n\} (n \geq 0)\) that satisfies \(a_0 = 0\), \(a_1 = 1\), and for all positive integers \(n\), \(a_{n+1} = 2a_n + 2007a_{n-1}\), find the smallest positive integer \(n\) such that \(2008\) divides \(a_n\).	2008	0.5
The figure \( A B C D E F \) is a regular hexagon. Evaluate the quotient $$ \frac{\text { Area of hexagon } A B C D E F}{\text { Area of triangle } A C D}. $$	3	0.5
Points $A, B$, and $C$ lie in that order on line $\ell$, such that $AB = 3$ and $BC = 2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.	\sqrt{10}	0.625
Find the distance between the face diagonal \( A_{1}B \) and the face diagonal \( B_{1}D_{1} \) in a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with a side length of 1.	\frac{\sqrt{3}}{3}	0.625
Howard chooses \( n \) different numbers from the list \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}, so that no two of his choices add up to a square. What is the largest possible value of \( n \)?	7	0.75
Given \( x, y, z \in \mathbf{Z} \) such that \(\left\{\begin{array}{l}x+ y + z = 3 \\ x^3 + y^3 + z^3 = 3\end{array}\right.\), determine the set of all possible values of \( x^2 + y^2 + z^2 \).	\{3, 57\}	0.75
In a certain school, there are 100 students in the second year of high school who are excellent in at least one of the three subjects: mathematics, physics, or chemistry. Among them, 70 students are excellent in mathematics, 65 in physics, and 75 in chemistry. There are 40 students who are excellent in both mathematics and physics, 45 in both mathematics and chemistry, and 25 students who are excellent in all three subjects. How many students are excellent in both physics and chemistry but not in mathematics?	25	0.875
Let \(\mathbb{Z}_{>0}\) be the set of positive integers. Find all functions \(f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}\) such that $$ m^{2}+f(n) \mid m f(m)+n $$ for all positive integers \(m\) and \(n\).	f(n) = n	0.75
There are 158 children standing in a line. Starting from the first child on the left, every other child receives an apple (the first child receives an apple). Starting from the first child on the right, every third child receives a banana (the first child receives a banana). Find the number of children who do not receive any fruit.	52	0.25
The entire surface of a cube with dimensions $13 \times 13 \times 13$ was painted red, and then this cube was cut into $1 \times 1 \times 1$ cubes. All faces of the $1 \times 1 \times 1$ cubes that were not painted red were painted blue. By what factor is the total area of the blue faces greater than the total area of the red faces?	12	0.875
How many positive integers less than 1998 are relatively prime to 1547? (Two integers are relatively prime if they have no common factors besides 1.)	1487	0.375
Given a triangle \( \triangle ABC \) with circumradius \( R \), it is known that \[ \frac{a \cos \alpha + b \cos \beta + c \cos \gamma}{a \sin \beta + b \sin \gamma + c \sin \alpha} = \frac{a + b + c}{9R}, \] where \( a, b, c \) are the lengths of the sides of \( \triangle ABC \), and \( \alpha, \beta, \gamma \) are the measures of angles \( \angle A, \angle B, \angle C \) respectively. Determine \( \alpha, \beta, \gamma \).	60^\circ	0.25
Given that point \( P \) moves on the circle \( x^{2} + (y - 4)^{2} = 1 \) and point \( Q \) moves on the ellipse \( \frac{x^{2}}{9} + y^{2} = 1 \), find the maximum value of \( \|PQ\| \).	3\sqrt{3} + 1	0.25
Let \( a < b < c < d \) be odd natural numbers such that \( ad = bc \) and \( a + d \) and \( b + c \) are powers of 2. Show that \( a = 1 \).	a=1	0.875
The series below includes the consecutive even integers from 2 to 2022 inclusive, where the signs of the terms alternate between positive and negative: $$ S=2-4+6-8+10-\cdots-2016+2018-2020+2022 $$ What is the value of $S$?	1012	0.625
Compute the limit of the function: $$\lim _{x \rightarrow 1} \frac{\cos (2 \pi x)}{2+\left(e^{\sqrt{x-1}}-1\right) \operatorname{arctg} \frac{x+2}{x-1}}$$	\frac{1}{2}	0.875
All three-digit natural numbers, whose first digits are odd and greater than 1, are written on the board. What is the maximum number of quadratic equations of the form \( a x^{2} + b x + c = 0 \) that can be created by using these numbers as \( a \), \( b \), and \( c \), each no more than once, such that all these equations have roots?	100	0.125
Given \( a_{n}=\log _{n}(n+1) \), determine \( \sum_{n=2}^{1023} \frac{1}{\log _{a_{n}} 100}=\frac{q}{p} \), where \( p \) and \( q \) are positive integers such that \( (p, q)=1 \). Find \( p + q \).	3	0.625
There are points \( A, B, C, D \) marked on a piece of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between any two given points in centimeters; b) compare two given numbers. What is the minimum number of operations needed for this device to definitively determine whether the quadrilateral \( ABCD \) is a rectangle?	9	0.375
In how many different ways can 1,000,000 be represented as the product of three natural numbers? Products that differ only in the order of the factors are considered the same.	139	0.5
Consider an equilateral triangle \(ABC\), where \(AB = BC = CA = 2011\). Let \(P\) be a point inside \(\triangle ABC\). Draw line segments passing through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Suppose \(DE : FG : HI = 8 : 7 : 10\). Find \(DE + FG + HI\).	4022	0.625
How many ordered pairs \((x, y)\) of positive integers, where \(x < y\), satisfy the equation $$ \frac{1}{x} + \frac{1}{y} = \frac{1}{2007}? $$	7	0.625
As shown in the figure, in parallelogram $ABCD$, the ratio $\frac{AE}{ED}=\frac{9}{5}$ and the ratio $\frac{BF}{FC}=\frac{7}{4}$. Find the ratio of the areas of $\triangle ACE$ and $\triangle BDF$.	\frac{99}{98}	0.75
Given \( f(x) = \lg (x+1) - \frac{1}{2} \log_{3} x \). (1) Solve the equation \( f(x) = 0 \); (2) Determine the number of subsets of the set \( M = \{ n \mid f(n^2 - 214n - 1998) \geq 0, n \in \mathbf{Z} \} \).	4	0.75
In a quiz, no two people had the same score and the score of each participant is equal to \( n + 2 - 2k \) where \( n \) is a constant and \( k \) is the rank of the participant. If the total score of all participants is 2009, find the smallest possible value of \( n \).	89	0.5
Find \( k \) such that, for all \( n \), the following expression is a perfect square: $$ 4 n^{2} + k n + 9 $$	12	0.5
Draw a rectangle on each side of a given rectangle such that the height of each new rectangle is one $n$-th of the length of the corresponding side of the original rectangle. Starting with rectangles of equal perimeter, can you choose a value of $n$ such that the area of the resulting shape, which consists of the original rectangle and the four added rectangles, is always the same?	4	0.5
In some 16 cells of an $8 \times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration?	16	0.625
In 12 days, it will be the year 2016. Haohau remarked: Up to now, I have only experienced 2 leap years, and the year I was born is a multiple of 9. How old will Haohau be in 2016?	9	0.75
How many ways are there to color every integer either red or blue such that \( n \) and \( n+7 \) are the same color for all integers \( n \), and there does not exist an integer \( k \) such that \( k, k+1 \), and \( 2k \) are all the same color?	6	0.25
A plane is drawn through each face of a cube. Into how many parts do these planes divide the space?	27	0.375
On the coordinate plane, the points $A(9, 1)$, $B(2, 0)$, $D(1, 5)$, and $E(9, 7)$ are given. Find the area of the pentagon $ABCDE$, where $C$ is the intersection point of the lines $AD$ and $BE$.	33.00	0.875
Let \(ABCD\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(AB\) and \(AD\), respectively, so that the area of square \(ABCD\) equals the area of triangle \(AEF\). If \(EF\) intersects \(BC\) at \(X\) and \(BX=6\), determine \(DF\).	\sqrt{13}	0.875
If \( x \) and \( y \) are both greater than 0, and \( x + 2y = 1 \), then what is the minimum value of \( u = \left( x + \frac{1}{x} \right) \left( y + \frac{1}{4y} \right) \)?	\frac{25}{8}	0.625
Let points \( C \) and \( D \) be the trisection points of \( AB \). At 8:00, person \( A \) starts walking from \( A \) to \( B \) at a constant speed. At 8:12, person \( B \) starts walking from \( B \) to \( A \) at a constant speed. After a few more minutes, person \( C \) starts walking from \( B \) to \( A \) at a constant speed. When \( A \) and \( B \) meet at point \( C \), person \( C \) reaches point \( D \). When \( A \) and \( C \) meet at 8:30, person \( B \) reaches \( A \) exactly at that time. At what time did person \( C \) start walking?	8:16	0.75
A column of cars, moving uniformly at the same speed, has a length of $5 \mathrm{km}$. In the last car is the head of the column, next to whom is a motorcyclist. On the orders of the leader, the motorcyclist increased his speed, caught up with the front car, delivered a package, immediately turned around and, with the same speed with which he went forward, went back to his place. While the motorcyclist completed the task, the column moved forward by 5 km. How many kilometers did the motorcyclist travel?	5(1 + \sqrt{2})	0.75
In November, Panteleimon and Gerasim each received 20 grades. Panteleimon received as many fives as Gerasim received fours, as many fours as Gerasim received threes, as many threes as Gerasim received twos, and as many twos as Gerasim received fives. Additionally, their average grades for November are the same. How many twos did Panteleimon receive in November?	5	0.625
What is the maximum integer number of liters of water that can be heated to boiling temperature using the heat obtained from burning solid fuel, if during the first 5 minutes of burning the fuel produces 480 kJ, and during each subsequent five-minute period $25\%$ less than during the previous one. The initial temperature of the water is $20^{\circ} \mathrm{C}$, the boiling temperature is $100^{\circ} \mathrm{C}$, and the specific heat capacity of water is 4.2 kJ.	5	0.625
In cyclic pentagon \(ABCDE\), \(\angle ABD = 90^\circ\), \(BC = CD\), and \(AE\) is parallel to \(BC\). If \(AB = 8\) and \(BD = 6\), find \(AE^2\).	\frac{338}{5}	0.125
Given a rectangle with dimensions \(100 \times 101\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.	200	0.5
How many paths are there on an $n \times n$ grid to go from point $(0, 0)$ to point $(n, n)$ using only moves of 1 up or 1 to the right?	\binom{2n}{n}	0.75
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the angle bisector of \(\angle D\).	10^\circ	0.75
Given that \( w>0 \) and that \( w-\frac{1}{w}=5 \), find the value of \( \left(w+\frac{1}{w}\right)^{2} \).	29	0.75
Given the sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=\log _{3}\left(1+\frac{2}{n^{2}+3 n}\right)$, find $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)$.	1	0.75
Find the coordinates of the center of gravity of a homogeneous plane figure bounded by the curves \( y = \frac{1}{2} x^2 \) and \( y = 2 \).	(0, 1.2)	0.5
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31, respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).	1705	0.75
Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?	505	0.625
Fill the six numbers $1, 3, 5, 7, 9, 11$ into the circles in the given diagram (each circle contains one number) so that the sum of the three numbers on each side equals 19. Then, find the sum of the three numbers in the circles that form the vertices of the triangle.	21	0.375
In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table?	80	0.5
Compute the limit of the function: \[ \lim _{x \rightarrow 0}\left(\frac{1 + x \cdot 2^{x}}{1 + x \cdot 3^{x}}\right)^{\frac{1}{x^{2}}} \]	\frac{2}{3}	0.875
At a sumo wrestling tournament, 20 sumo wrestlers participated. After weighing, it was found that the average weight of the wrestlers is 125 kg. What is the maximum possible number of wrestlers weighing more than 131 kg, given that according to the rules, individuals weighing less than 90 kg cannot participate in sumo wrestling?	17	0.625
In the following maze, each of the dashed segments is randomly colored either black or white. What is the probability that there will exist a path from side \( A \) to side \( B \) that does not cross any of the black lines? ![Maze](https://cdn.mathpix.com/cropped/2024_04_17_a143cf101bf1bb453a87g-2.jpg?height=564&width=1134&top_left_y=580&top_left_x=488)	\frac{1}{2}	0.375
Determine the number of quadruples of natural numbers \((a, b, c, d)\) satisfying \(a \cdot b \cdot c \cdot d = 98\). Note: \((98, 1, 1, 1)\) and \((1, 1, 98, 1)\) are considered different quadruples.	40	0.875
How many five-digit numbers are roots of the equation \( x = [\sqrt{x} + 1][\sqrt{x}] \)? The symbol \([a]\) denotes the integer part of the number \(a\), which is the largest integer not exceeding \(a\).	216	0.875
If a sequence of numbers \(a_{1}, a_{2}, \cdots\) satisfies, for any positive integer \(n\), $$ a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2}, $$ then what is the value of \(a_{1} a_{2} \cdots a_{2016}\)?	2016\sqrt{2} - 2015	0.375
Problems A, B, and C were posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as solved C. The number solving only A was one more than the number solving A and at least one other. The number solving just A equalled the number solving just B plus the number solving just C. How many solved just B?	6	0.75

The three-digit number ends with the digit 2. If this digit is moved to the beginning of the number, the resulting number will be 18 greater than the original number. Find this number.

202

0.875

At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?

49

0.5

One ball is taken at random from a bag containing $(b-4)$ white balls and $(b+46)$ red balls. If $\frac{c}{6}$ is the probability that the ball is white, find $c$.

2

0.625

In a certain book, there were 100 statements written as follows: 1) "In this book, there is exactly one false statement." 2) "In this book, there are exactly two false statements." ... 3) "In this book, there are exactly one hundred false statements." Which of these statements is true?

99

0.875

A mouse is gnawing on a cube of cheese with an edge of 3, divided into 27 unit cubes. When the mouse eats a cube, it moves to another cube that shares a face with the previous one. Can the mouse eat the entire cube except for the central unit cube?

\text{No}

0.75

The side of an equilateral triangle inscribed in a circle is $a$. Calculate the area of the square inscribed in the same circle.

\frac{2 a^2}{3}

0.875

The product of the midline of a trapezoid and the segment connecting the midpoints of its diagonals equals 25. Find the area of the trapezoid if its height is three times the difference of its bases.

150

0.875

A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different.

16

0.375

If $ 16^{\sin^2 x} + 16^{\cos^2 x} = 10 $, then $ \cos 4x = $ _______.

-\frac{1}{2}

0.75

In a single round-robin table tennis tournament, a) 10 players, b) 11 players competed. What is the minimum number of matches the champion must have won, given that there was no tie for first place?

6

0.625

There exists $ x_{0} < 0 $ such that $ x^{2} + |x - a| - 2 < 0 $ (where $ a \in \mathbb{Z} $) is always true. Find the sum of all values of $ a $ that satisfy this condition.

-2

0.75

As shown in the figure, point $D$ is an interior point of $\triangle ABC$. $BD$ intersects $AC$ at $E$, and $CD$ intersects $AB$ at $F$, with $AF = BF = CD$ and $CE = DE$. Find the measure of $\angle BFC$.

120^\circ

0.625

Calculate the volume of a tetrahedron with vertices at points $ A_{1}, A_{2}, A_{3}, A_{4} $ and its height from vertex $ A_{4} $ to the face $ A_{1}A_{2}A_{3} $. $ A_{1}(3, 10, -1) $ $ A_{2}(-2, 3, -5) $ $ A_{3}(-6, 0, -3) $ $ A_{4}(1, -1, 2) $

7

0.375

Let $ P $ be a polynomial of degree 4 such that $ P(0) = P(1) = 1 $, $ P(2) = 4 $, $ P(3) = 9 $, and $ P(4) = 16 $. Calculate $ P(-2) $.

19

0.875

For $x, y, z \geq 1$, find the minimum value of the expression $$ A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x} $$

1

0.875

A town has $ c $ celebrities and $ m $ weekly magazines. One week each celebrity was mentioned in an odd number of magazines, and each magazine mentioned an odd number of celebrities. Show that $ m $ and $ c $ are both even or both odd. How many ways are there for each magazine to mention an odd number of celebrities so that each celebrity is mentioned an odd number of times?

2^{(m-1)(c-1)}

0.625

Find the rank of the $ (2n+1) \times (2n+1) $ skew-symmetric matrix with entries given by: - $ a_{ij} = 1 $ for $ i - j = -2n, -(2n-1), \ldots, -(n+1) $ - $ a_{ij} = -1 $ for $ i - j = -n, -(n-1), \ldots, -1 $ - $ a_{ij} = 1 $ for $ i - j = 1, 2, \ldots, n $ - $ a_{ij} = -1 $ for $ i - j = n+1, n+2, \ldots, 2n+1 $ In other words, the main diagonal is zeros, the diagonals immediately below the main diagonal (up to $ n $ diagonals) are $ 1s $, the diagonals below those are $ -1s $, the diagonals immediately above the main diagonal (up to $ n $ diagonals) are $ -1s $, and the diagonals above those are $ 1s $.

2n

0.75

Is it possible to place 8 different numbers, none of which are divisible by 13, from the range of 1 to 245 at the vertices of a cube such that the numbers in adjacent vertices have a common divisor greater than 1, while the numbers in non-adjacent vertices do not share any common divisor greater than 1?

\text{No}

0.875

Let $ x $ be a non-zero real number such that \[ \sqrt[5]{x^{3}+20 x}=\sqrt[3]{x^{5}-20 x} \]. Find the product of all possible values of $ x $.

-5

0.125

The manager of a store checked the sales price in 2006 of a television from the brand VejoTudo. He found a partially faded invoice that read: "lot of 72 VejoTudo TVs sold for R$ \ldots 679 \ldots reais," where the digits of the unit and the ten thousand place were illegible. What was the sales price in 2006 of each of those televisions?

511

0.625

$ N $ is a 5-digit number formed by 5 different non-zero digits, and $ N $ is equal to the sum of all 3-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $ N $.

35964

0.75

Suppose that $ x - y = 1 $. Find the value of $ x^4 - xy^3 - x^3 y - 3x^2 y + 3xy^2 + y^4 $.

1

0.875

Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient?

231

0.125

Given a sequence of positive integers $\left\{a_{n}\right\}$ defined by $a_{0}=m$ and $a_{n+1}=a_{n}^{5}+487$ for $n \in \mathbf{N}$, find the value of $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares.

9

0.75

Find the smallest positive integer that can be expressed as the sum of 9 consecutive integers, 10 consecutive integers, and 11 consecutive integers.

495

0.625

Each cell in a $5 \times 5$ grid contains a natural number written in invisible ink. It is known that the sum of all the numbers is 200, and the sum of any three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the grid?

16

0.25

Masha was given a box with differently colored beads (each bead having a unique color, with a total of $\mathrm{n}$ beads in the box). Masha chose seven beads for her dress and decided to try out all possible combinations of these beads on the dress (thus, Masha considers sewing one, two, three, four, five, six, or seven beads, with the order of the beads not mattering to her). Then she counted the total number of combinations and was very surprised that the number turned out to be odd. 1) What number did Masha get? 2) Is it true that when selecting from a set with an even number of beads, Masha could get an even number of combinations? 3) Is it true that if the order of the beads sewn onto the dress mattered to Masha, she could have ended up with either an even or odd number of combinations?

\text{Yes}

0.125

Let $ a $ be a strictly positive integer. Suppose that $ 4(a^n + 1) $ is the cube of an integer for every positive integer $ n $. Find $ a $.

1

0.75

January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year?

52

0.75

Given a right triangle $ \triangle ABC $ with vertices $ A, B, $ and $ C $ on the parabola $ y = x^2 $, and the hypotenuse $ AB $ parallel to the x-axis, find the length of the altitude from $ C $ to the hypotenuse $ AB $.

1

0.875

Find the smallest natural number $ n $ for which the number $ A = n^3 + 12n^2 + 15n + 180 $ is divisible by 23.

10

0.375

What is the smallest number of participants that could have been in the school drama club if the fifth graders made up more than $22\%$ but less than $27\%$ of the club, the sixth graders made up more than $25\%$ but less than $35\%$, and the seventh graders made up more than $35\%$ but less than $45\%$ (there were no participants from other grades)?

9

0.25

Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, ..., 2014) until we obtain a value less than or equal to the previous roll. Let E be the expected number of times we roll the die. Find the nearest integer to 100E.

272

0.375

If the complex number $ z $ satisfies \[ |z-1| + |z-3-2i| = 2\sqrt{2}, \] then the minimum value of $ |z| $ is ______ .

1

0.875

Find the equation of the normal line to the given curve at the point with the abscissa $x_{0}$. $y=\sqrt{x}-3 \sqrt[3]{x}, x_{0}=64$

x = 64

0.875

Given a cube with a side length of 4, is it possible to entirely cover three of its faces, which share a common vertex, using 16 paper rectangular strips each measuring $1 \times 3$?

\text{No}

0.625

Compute the limit of the numerical sequence: \[ \lim _{n \rightarrow \infty} \frac{\sqrt{n+1}-\sqrt[3]{n^{3}+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^{5}+1}} \]

1

0.75

Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers?

1665

0.125

Let $a$ and $b$ be coprime integers. Show that $a + b$ and $ab$ are coprime.

1

0.625

What number must be added to both terms of a fraction to obtain the reciprocal of that same fraction?

- (a + b)

0.875

Yvan and Zoé play the following game. Let $ n \in \mathbb{N} $. The integers from 1 to $ n $ are written on $ n $ cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards. What is the smallest value of $ n $ for which Zoé can ensure that she can play her two turns?

14

0.25

Two cars, Car A and Car B, simultaneously depart from point $A$ to point $B$. At the start, Car A’s speed is 2.5 km/h faster than Car B’s speed. After 10 minutes, Car A slows down; another 5 minutes later, Car B also slows down, and at this time, Car B is 0.5 km/h slower than Car A. After another 25 minutes, both cars arrive at point $B$ at the same time. By how many kilometers per hour did Car A's speed decrease?

10 \text{ km/h}

0.625

Place a known mathematical symbol between the numbers 4 and 5 such that the result is a number greater than 4 but less than 5.

4.5

0.625

The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM?

30 \text{ minutes}

0.75

Given a sequence $\{a_n\} (n \geq 0)$ that satisfies $a_0 = 0$, $a_1 = 1$, and for all positive integers $n$, $a_{n+1} = 2a_n + 2007a_{n-1}$, find the smallest positive integer $n$ such that $2008$ divides $a_n$.

2008

0.5

The figure $ A B C D E F $ is a regular hexagon. Evaluate the quotient $$ \frac{\text { Area of hexagon } A B C D E F}{\text { Area of triangle } A C D}. $$

3

0.5

Points $A, B$, and $C$ lie in that order on line $\ell$, such that $AB = 3$ and $BC = 2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible.

\sqrt{10}

0.625

Find the distance between the face diagonal $ A_{1}B $ and the face diagonal $ B_{1}D_{1} $ in a cube $ ABCD-A_{1}B_{1}C_{1}D_{1} $ with a side length of 1.

\frac{\sqrt{3}}{3}

0.625

Howard chooses $ n $ different numbers from the list \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}, so that no two of his choices add up to a square. What is the largest possible value of $ n $?

7

0.75

Given $ x, y, z \in \mathbf{Z} $ such that $\left\{\begin{array}{l}x+ y + z = 3 \\ x^3 + y^3 + z^3 = 3\end{array}\right.$, determine the set of all possible values of $ x^2 + y^2 + z^2 $.

\{3, 57\}

0.75

In a certain school, there are 100 students in the second year of high school who are excellent in at least one of the three subjects: mathematics, physics, or chemistry. Among them, 70 students are excellent in mathematics, 65 in physics, and 75 in chemistry. There are 40 students who are excellent in both mathematics and physics, 45 in both mathematics and chemistry, and 25 students who are excellent in all three subjects. How many students are excellent in both physics and chemistry but not in mathematics?

25

0.875

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $$ m^{2}+f(n) \mid m f(m)+n $$ for all positive integers $m$ and $n$.

f(n) = n

0.75

There are 158 children standing in a line. Starting from the first child on the left, every other child receives an apple (the first child receives an apple). Starting from the first child on the right, every third child receives a banana (the first child receives a banana). Find the number of children who do not receive any fruit.

52

0.25

The entire surface of a cube with dimensions $13 \times 13 \times 13$ was painted red, and then this cube was cut into $1 \times 1 \times 1$ cubes. All faces of the $1 \times 1 \times 1$ cubes that were not painted red were painted blue. By what factor is the total area of the blue faces greater than the total area of the red faces?

12

0.875

How many positive integers less than 1998 are relatively prime to 1547? (Two integers are relatively prime if they have no common factors besides 1.)

1487

0.375

Given a triangle $ \triangle ABC $ with circumradius $ R $, it is known that \[ \frac{a \cos \alpha + b \cos \beta + c \cos \gamma}{a \sin \beta + b \sin \gamma + c \sin \alpha} = \frac{a + b + c}{9R}, \] where $ a, b, c $ are the lengths of the sides of $ \triangle ABC $, and $ \alpha, \beta, \gamma $ are the measures of angles $ \angle A, \angle B, \angle C $ respectively. Determine $ \alpha, \beta, \gamma $.

60^\circ

0.25

Given that point $ P $ moves on the circle $ x^{2} + (y - 4)^{2} = 1 $ and point $ Q $ moves on the ellipse $ \frac{x^{2}}{9} + y^{2} = 1 $, find the maximum value of $ |PQ| $.

3\sqrt{3} + 1

0.25

Let $ a < b < c < d $ be odd natural numbers such that $ ad = bc $ and $ a + d $ and $ b + c $ are powers of 2. Show that $ a = 1 $.

a=1

0.875

The series below includes the consecutive even integers from 2 to 2022 inclusive, where the signs of the terms alternate between positive and negative: $$ S=2-4+6-8+10-\cdots-2016+2018-2020+2022 $$ What is the value of $S$?

1012

0.625

Compute the limit of the function: $$\lim _{x \rightarrow 1} \frac{\cos (2 \pi x)}{2+\left(e^{\sqrt{x-1}}-1\right) \operatorname{arctg} \frac{x+2}{x-1}}$$

\frac{1}{2}

0.875

All three-digit natural numbers, whose first digits are odd and greater than 1, are written on the board. What is the maximum number of quadratic equations of the form $ a x^{2} + b x + c = 0 $ that can be created by using these numbers as $ a $, $ b $, and $ c $, each no more than once, such that all these equations have roots?

100

0.125

Given $ a_{n}=\log _{n}(n+1) $, determine $ \sum_{n=2}^{1023} \frac{1}{\log _{a_{n}} 100}=\frac{q}{p} $, where $ p $ and $ q $ are positive integers such that $ (p, q)=1 $. Find $ p + q $.

3

0.625

There are points $ A, B, C, D $ marked on a piece of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between any two given points in centimeters; b) compare two given numbers. What is the minimum number of operations needed for this device to definitively determine whether the quadrilateral $ ABCD $ is a rectangle?

9

0.375

In how many different ways can 1,000,000 be represented as the product of three natural numbers? Products that differ only in the order of the factors are considered the same.

139

0.5

Consider an equilateral triangle $ABC$, where $AB = BC = CA = 2011$. Let $P$ be a point inside $\triangle ABC$. Draw line segments passing through $P$ such that $DE \parallel BC$, $FG \parallel CA$, and $HI \parallel AB$. Suppose $DE : FG : HI = 8 : 7 : 10$. Find $DE + FG + HI$.

4022

0.625

How many ordered pairs $(x, y)$ of positive integers, where $x < y$, satisfy the equation $$ \frac{1}{x} + \frac{1}{y} = \frac{1}{2007}? $$

7

0.625

As shown in the figure, in parallelogram $ABCD$, the ratio $\frac{AE}{ED}=\frac{9}{5}$ and the ratio $\frac{BF}{FC}=\frac{7}{4}$. Find the ratio of the areas of $\triangle ACE$ and $\triangle BDF$.

\frac{99}{98}

0.75

Given $ f(x) = \lg (x+1) - \frac{1}{2} \log_{3} x $. (1) Solve the equation $ f(x) = 0 $; (2) Determine the number of subsets of the set $ M = \{ n \mid f(n^2 - 214n - 1998) \geq 0, n \in \mathbf{Z} \} $.

4

0.75

In a quiz, no two people had the same score and the score of each participant is equal to $ n + 2 - 2k $ where $ n $ is a constant and $ k $ is the rank of the participant. If the total score of all participants is 2009, find the smallest possible value of $ n $.

89

0.5

Find $ k $ such that, for all $ n $, the following expression is a perfect square: $$ 4 n^{2} + k n + 9 $$

12

0.5

Draw a rectangle on each side of a given rectangle such that the height of each new rectangle is one $n$-th of the length of the corresponding side of the original rectangle. Starting with rectangles of equal perimeter, can you choose a value of $n$ such that the area of the resulting shape, which consists of the original rectangle and the four added rectangles, is always the same?

4

0.5

In some 16 cells of an $8 \times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration?

16

0.625

In 12 days, it will be the year 2016. Haohau remarked: Up to now, I have only experienced 2 leap years, and the year I was born is a multiple of 9. How old will Haohau be in 2016?

9

0.75

How many ways are there to color every integer either red or blue such that $ n $ and $ n+7 $ are the same color for all integers $ n $, and there does not exist an integer $ k $ such that $ k, k+1 $, and $ 2k $ are all the same color?

6

0.25

A plane is drawn through each face of a cube. Into how many parts do these planes divide the space?

27

0.375

On the coordinate plane, the points $A(9, 1)$, $B(2, 0)$, $D(1, 5)$, and $E(9, 7)$ are given. Find the area of the pentagon $ABCDE$, where $C$ is the intersection point of the lines $AD$ and $BE$.

33.00

0.875

Let $ABCD$ be a square of side length 13. Let $E$ and $F$ be points on rays $AB$ and $AD$, respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX=6$, determine $DF$.

\sqrt{13}

0.875

If $ x $ and $ y $ are both greater than 0, and $ x + 2y = 1 $, then what is the minimum value of $ u = \left( x + \frac{1}{x} \right) \left( y + \frac{1}{4y} \right) $?

\frac{25}{8}

0.625

Let points $ C $ and $ D $ be the trisection points of $ AB $. At 8:00, person $ A $ starts walking from $ A $ to $ B $ at a constant speed. At 8:12, person $ B $ starts walking from $ B $ to $ A $ at a constant speed. After a few more minutes, person $ C $ starts walking from $ B $ to $ A $ at a constant speed. When $ A $ and $ B $ meet at point $ C $, person $ C $ reaches point $ D $. When $ A $ and $ C $ meet at 8:30, person $ B $ reaches $ A $ exactly at that time. At what time did person $ C $ start walking?

8:16

0.75

A column of cars, moving uniformly at the same speed, has a length of $5 \mathrm{km}$. In the last car is the head of the column, next to whom is a motorcyclist. On the orders of the leader, the motorcyclist increased his speed, caught up with the front car, delivered a package, immediately turned around and, with the same speed with which he went forward, went back to his place. While the motorcyclist completed the task, the column moved forward by 5 km. How many kilometers did the motorcyclist travel?

5(1 + \sqrt{2})

0.75

In November, Panteleimon and Gerasim each received 20 grades. Panteleimon received as many fives as Gerasim received fours, as many fours as Gerasim received threes, as many threes as Gerasim received twos, and as many twos as Gerasim received fives. Additionally, their average grades for November are the same. How many twos did Panteleimon receive in November?

5

0.625

What is the maximum integer number of liters of water that can be heated to boiling temperature using the heat obtained from burning solid fuel, if during the first 5 minutes of burning the fuel produces 480 kJ, and during each subsequent five-minute period $25\%$ less than during the previous one. The initial temperature of the water is $20^{\circ} \mathrm{C}$, the boiling temperature is $100^{\circ} \mathrm{C}$, and the specific heat capacity of water is 4.2 kJ.

5

0.625

In cyclic pentagon $ABCDE$, $\angle ABD = 90^\circ$, $BC = CD$, and $AE$ is parallel to $BC$. If $AB = 8$ and $BD = 6$, find $AE^2$.

\frac{338}{5}

0.125

Given a rectangle with dimensions $100 \times 101$, divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.

200

0.5

How many paths are there on an $n \times n$ grid to go from point $(0, 0)$ to point $(n, n)$ using only moves of 1 up or 1 to the right?

\binom{2n}{n}

0.75

Given a trapezoid $ABCD$ with bases $AB$ and $CD$, and angles $\angle C = 30^\circ$ and $\angle D = 80^\circ$. Find $\angle ACB$, given that $DB$ is the angle bisector of $\angle D$.

10^\circ

0.75

Given that $ w>0 $ and that $ w-\frac{1}{w}=5 $, find the value of $ \left(w+\frac{1}{w}\right)^{2} $.

29

0.75

Given the sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=\log _{3}\left(1+\frac{2}{n^{2}+3 n}\right)$, find $\lim _{n \rightarrow \infty}\left(a_{1}+a_{2}+\cdots+a_{n}\right)$.

1

0.75

Find the coordinates of the center of gravity of a homogeneous plane figure bounded by the curves $ y = \frac{1}{2} x^2 $ and $ y = 2 $.

(0, 1.2)

0.5

The bases $AB$ and $CD$ of trapezoid $ABCD$ are 55 and 31, respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.

1705

0.75

Let $ A $ be a subset of $\{1, 2, 3, \ldots, 2019\}$ having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in $ A $?

505

0.625

Fill the six numbers $1, 3, 5, 7, 9, 11$ into the circles in the given diagram (each circle contains one number) so that the sum of the three numbers on each side equals 19. Then, find the sum of the three numbers in the circles that form the vertices of the triangle.

21

0.375

In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table?

80

0.5

Compute the limit of the function: \[ \lim _{x \rightarrow 0}\left(\frac{1 + x \cdot 2^{x}}{1 + x \cdot 3^{x}}\right)^{\frac{1}{x^{2}}} \]

\frac{2}{3}

0.875

At a sumo wrestling tournament, 20 sumo wrestlers participated. After weighing, it was found that the average weight of the wrestlers is 125 kg. What is the maximum possible number of wrestlers weighing more than 131 kg, given that according to the rules, individuals weighing less than 90 kg cannot participate in sumo wrestling?

17

0.625

In the following maze, each of the dashed segments is randomly colored either black or white. What is the probability that there will exist a path from side $ A $ to side $ B $ that does not cross any of the black lines? ![Maze](https://cdn.mathpix.com/cropped/2024_04_17_a143cf101bf1bb453a87g-2.jpg?height=564&width=1134&top_left_y=580&top_left_x=488)

\frac{1}{2}

0.375

Determine the number of quadruples of natural numbers $(a, b, c, d)$ satisfying $a \cdot b \cdot c \cdot d = 98$. Note: $(98, 1, 1, 1)$ and $(1, 1, 98, 1)$ are considered different quadruples.

40

0.875

How many five-digit numbers are roots of the equation $ x = [\sqrt{x} + 1][\sqrt{x}] $? The symbol $[a]$ denotes the integer part of the number $a$, which is the largest integer not exceeding $a$.

216

0.875

If a sequence of numbers $a_{1}, a_{2}, \cdots$ satisfies, for any positive integer $n$, $$ a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2}, $$ then what is the value of $a_{1} a_{2} \cdots a_{2016}$?

2016\sqrt{2} - 2015

0.375

Problems A, B, and C were posed in a mathematical contest. 25 competitors solved at least one of the three. Amongst those who did not solve A, twice as many solved B as solved C. The number solving only A was one more than the number solving A and at least one other. The number solving just A equalled the number solving just B plus the number solving just C. How many solved just B?

6

0.75