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A group of $n$ friends takes $r$ distinct photos (no two photos have exactly the same people) each containing at least one person. Find the maximum $r$ such that each person appears at most once. | r = n | 0.875 |
In Figure 1, if the sum of the interior angles is \(a^{\circ}\), find \(a\).
If the \(n^{\text{th}}\) term of the arithmetic progression \(80, 130, 180, 230, 280, \ldots\) is \(a\), find \(n\).
In Figure 2, \(AP: PB = 2: 1\).
If \(AC = 33 \text{ cm}\), \(BD = n \text{ cm}\), \(PQ = x \text{ cm}\), find \(x\).
If \(K = \frac{\sin 65^{\circ} \tan ^{2} 60^{\circ}}{\tan 30^{\circ} \cos 30^{\circ} \cos x^{\circ}}\), find \(K\). | 6 | 0.5 |
At a certain meeting, a total of \(12k\) people attended. Each person greeted exactly \(3k+6\) other people. For any two individuals, the number of people who greeted both of them is the same. How many people attended the meeting? | 36 | 0.625 |
Petya wrote down a sequence of ten natural numbers as follows: he wrote the first two numbers randomly, and each subsequent number, starting from the third, was equal to the sum of the two preceding numbers. Find the fourth number if the seventh number is 42 and the ninth number is 110. | 10 | 0.75 |
Given the sequence \(a_{n}=\sqrt{4+\frac{1}{n^{2}}}+\sqrt{4+\frac{2}{n^{2}}}+\cdots+\sqrt{4+\frac{n}{n^{2}}}-2 n\) where \(n\) is a positive integer, find the value of \(\lim _{n \rightarrow \infty} a_{n}\). | \frac{1}{8} | 0.75 |
Let \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) be a function that satisfies the following conditions:
1. \( f(1)=1 \)
2. \( f(2n)=f(n) \)
3. \( f(2n+1)=f(n)+1 \)
What is the greatest value of \( f(n) \) for \( 1 \leqslant n \leqslant 2018 \) ? | 10 | 0.375 |
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 a $6 \times 6$ toy board, each cell represents a light button. When someone presses a button, it lights up if it is off and turns off if it is on. Additionally, all buttons that share a side with the pressed button also change their state: from on to off or from off to on. Starting with all buttons off and pressing each button on the board exactly once, one at a time and in any order, how many buttons will be lit at the end? | 20 | 0.625 |
The distance between points $A$ and $B$ is 500 kilometers. Two cyclists, Alpha and Beta, start from point $A$ to point $B$ at the same time. Alpha cycles 30 kilometers per day, and Beta cycles 50 kilometers per day but rests every other day. At the end of the $\qquad$ day, the remaining distance Beta has to travel to point $B$ is twice the remaining distance Alpha has to travel. | 15 | 0.5 |
The roots of the quadratic equation \( ax^2 + bx + c = 0 \), which has integer coefficients, are distinct and both lie between 0 and 1. Show that if \( a \) is positive, it is at least 5. | 5 | 0.75 |
In a class of 50 students, each student has either a set square or a ruler. 28 students have rulers. Among those who have set squares, 14 are boys. Given that there are 31 girls in the class, how many girls have rulers? | 23 | 0.625 |
A rectangle with a perimeter of 100 cm is divided into 88 identical smaller rectangles by seven vertical and ten horizontal cuts. What is the perimeter of each? Additionally, the total length of all cuts is 434 cm. | 11 | 0.75 |
Let \( S_{n} \) be the sum of the elements of all 3-element subsets of the set \( A=\left\{1, \frac{1}{2}, \frac{1}{4}, \cdots, \frac{1}{2^{n}1}\right\} \). Evaluate \(\lim _{n \rightarrow \infty} \frac{S_{n}}{n^{2}}\). | 1 | 0.625 |
Given a positive integer \( n \) less than 2006, such that
\[
\left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{n}{6} \right\rfloor = \frac{n}{2}.
\]
Determine how many such \( n \) there are. | 334 | 0.625 |
Let \( F(x) = |f(x) \cdot g(x)| \), where \( f(x) = a x^{2} + b x + c \) and \( g(x) = c x^{2} + b x + a \), with \( x \in [-1, 1] \). For any \( a, b, c \) such that \( |f(x)| \leq 1 \), find the maximum value of \( F(x) \) as \( a, b, \) and \( c \) vary. | 2 | 0.5 |
It is known that the numbers EGGPLANT and TOAD are divisible by 3. What is the remainder when the number CLAN is divided by 3? (Letters denote digits, identical letters represent identical digits, different letters represent different digits). | 0 | 0.625 |
What is the maximum number of handshakes that could have occurred in a group of 20 people, given that no matter which three people are chosen from the group, there will always be at least two who have not shaken hands? | 100 | 0.875 |
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides? | 144 | 0.875 |
If \( x \) and \( y \) are integers that satisfy the equation \( y^{2} + 3x^{2}y^{2} = 30x^{2} + 517 \), find the value of \( 3x^{2}y^{2} \). | 588 | 0.625 |
In triangle \( \triangle ABC \), the side lengths are \( AC = 6 \), \( BC = 9 \), and \( \angle C = 120^\circ \). Find the length of the angle bisector \( CD \) of \( \angle C \). | \frac{18}{5} | 0.125 |
Find all positive integers $k$ such that for any positive numbers $a, b, c$ that satisfy the inequality $k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)$, there always exists a triangle with side lengths $a, b, c$. | 6 | 0.625 |
Non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{2016} \) and real numbers \( y_{1}, y_{2}, \cdots, y_{2016} \) satisfy:
(1) \( x_{k}^{2}+y_{k}^{2}=1 \) for \( k=1,2, \cdots, 2016 \);
(2) \( y_{1}+y_{2}+\cdots+y_{2016} \) is odd.
Find the minimum value of \( x_{1}+x_{2}+\cdots+x_{2016} \). | 1 | 0.75 |
12 Smurfs are sitting around a circular table, and each Smurf dislikes the 2 Smurfs adjacent to him but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must be made up of Smurfs who do not dislike one another. How many ways are there to form such a team? | 36 | 0.5 |
In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each tourist took no more than one photograph of each of the three monuments. It turned out that any two tourists together had photographs of all three monuments. What is the minimum number of photographs that all the tourists together could have taken? | 123 | 0.625 |
A teacher wrote down three positive integers on the whiteboard: 1125, 2925, and \( N \), and asked her class to compute the least common multiple of the three numbers. One student misread 1125 as 1725 and computed the least common multiple of 1725, 2925, and \( N \) instead. The answer he obtained was the same as the correct answer. Find the least possible value of \( N \). | 2875 | 0.5 |
In an airspace, there are clouds. It turned out that the space can be divided into parts by ten planes so that each part contains no more than one cloud. Through how many clouds could an airplane fly at most while following a straight course? | 11 | 0.75 |
Find the smallest natural number that is greater than the sum of its digits by 1755. | 1770 | 0.875 |
Find the sum of the digits of the product $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\cdots\left(1+\frac{1}{2006}\right)\left(1+\frac{1}{2007}\right)$. | 5 | 0.875 |
The teacher wrote the number 1818 on the board. Vasya noticed that if you insert a multiplication sign between the hundreds and tens places, the value of the resulting expression is a perfect square (18 × 18 = 324 = 18²). What is the next four-digit number after 1818 that has the same property? | 1832 | 0.375 |
Find the radius of the circumscribed circle around an isosceles trapezoid with bases of lengths 2 cm and 14 cm, and a lateral side of 10 cm. | 5\sqrt{2} | 0.625 |
In a right triangle \(PQR\), the leg \(PQ\) is equal to 12. The length of the angle bisector \(QS\) is \(6\sqrt{5}\). Find the hypotenuse \(QR\). | 20 | 0.5 |
Given five points inside an equilateral triangle of area 1, show that one can find three equilateral triangles with total area at most 0.64 and sides parallel to the original triangle, so that each of the five points is inside one or more of the new triangles. | 0.64 | 0.5 |
Given \( a_{1}, a_{2}, \cdots, a_{n} \) are real numbers such that \( a_{1} + a_{2} + \cdots + a_{n} = k \). Find the minimum value of \( a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \). | \frac{k^2}{n} | 0.75 |
Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of the road eleven times and Sal seven times. Find the number of times the bikers passed each other moving in opposite directions. | 8 | 0.125 |
Martin, Tomáš and Jirka guessed the result of the following expression:
$$
2.4 - 1.5 \cdot 3.6 \div 2 =
$$
Their average guess differed from the correct result by 0.4. Tomáš guessed $-1.2$ and Jirka guessed 1.7. Find the number that Martin guessed, knowing that the worst guess differed from the correct result by 2. | -0.2 | 0.875 |
The number \(a\) is a root of the quadratic equation \(x^{2} - x - 50 = 0\). Find the value of \(a^{3} - 51a\). | 50 | 0.75 |
If \( a \) is the smallest prime number which can divide the sum \( 3^{11} + 5^{13} \), find the value of \( a \). | 2 | 0.875 |
Find the sum of the squares of the real roots of the equation \(2x^{4} - 3x^{3} + 7x^{2} - 9x + 3 = 0\). | \frac{5}{4} | 0.75 |
The probability of event $A$ occurring in each trial is 1/2. Using Chebyshev's inequality, estimate the probability that the number $X$ of occurrences of event $A$ lies between 40 and 60, if 100 independent trials are conducted. | 0.75 | 0.875 |
The integers \( x, y, z \), and \( p \) satisfy the equation \( x^2 + xz - xy - yz = -p \). Given that \( p \) is prime, what is the value of \( |y+z| \) in terms of \( p \) ? Your answer should refer to the variable \( p \) so that it works for every prime number \( p \). | p+1 | 0.625 |
A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle. | 13 | 0.25 |
Find the smallest positive integer \( a \) such that the equation \( ax^2 - bx + c = 0 \) has two distinct roots in the interval \( 0 < x < 1 \) for some integers \( b \) and \( c \). | 5 | 0.625 |
Determine the smallest natural number \( n \) such that \( n! \) (n factorial) ends with exactly 1987 zeros. | 7960 | 0.125 |
Given a positive integer $n \geq 3$, if an $n$-vertex simple graph $G(V, E)$ is connected such that removing all the edges of any cycle in $G$ makes $G$ disconnected, find the maximum possible value of $|E(G)|$. | 2n - 3 | 0.375 |
A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left. | \frac{2}{3} | 0.875 |
In a given circle with radius \( R \), draw a regular pentagon \( ABCDE \). The perpendicular from vertex \( A \) to the opposite side \( DC \) intersects the circumference of the circle at \( A' \). Show that
$$
\overline{A^{\prime} B} - \overline{A^{\prime} C} = R.
$$ | A'B - A'C = R | 0.75 |
Find the set of pairs of real numbers \((x, y)\) that satisfy the conditions:
$$
\left\{
\begin{array}{l}
3^{-x} y^{4}-2 y^{2}+3^{x} \leq 0 \\
27^{x}+y^{4}-3^{x}-1=0
\end{array}
\right.
$$
Compute the values of the expression \(x_{k}^{3}+y_{k}^{3}\) for each solution \((x_{k}, y_{k})\) of the system and find the minimum among them. In the answer, specify the found minimum value, if necessary rounding it to two decimal places. If the original system has no solutions, write the digit 0 in the answer field. | -1 | 0.75 |
In an isosceles trapezoid, the midline is equal to \( a \), and the diagonals are perpendicular to each other. Find the area of the trapezoid. | a^2 | 0.625 |
Given that \( p(x) \) is a monic polynomial of degree 2003 with integer coefficients, show that the polynomial \( p(x)^2 - 25 \) cannot have more than 2003 distinct integer roots. | 2003 | 0.75 |
Find the maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$. | k_{\max} = 100 | 0.75 |
A seven-digit number \(\overline{m0A0B9C}\) is a multiple of 33. We denote the number of such seven-digit numbers as \(a_{m}\). For example, \(a_{5}\) represents the count of seven-digit numbers of the form \(\overline{50A0B9C}\) that are multiples of 33. Find the value of \(a_{2}-a_{3}\). | 8 | 0.5 |
For how many remaining years this century (after 2025 and up to and including 2099) will the highest common factor of the first two digits and the last two digits be equal to one? | 30 | 0.5 |
Let \(a_{1}, a_{2}, \ldots, a_{n}\) be a sequence of distinct positive integers such that \(a_{1} + a_{2} + \cdots + a_{n} = 2021\) and \(a_{1} a_{2} \cdots a_{n}\) is maximized. If \(M = a_{1} a_{2} \cdots a_{n}\), compute the largest positive integer \(k\) such that \(2^{k} \mid M\). | 62 | 0.125 |
On an island, there are 10 people. Some of them are truth-tellers and others are liars. Each person has thought of an integer. The first person says, "My number is greater than 1." The second person says, "My number is greater than 2." ... The tenth person says, "My number is greater than 10."
Afterwards, these ten people stand in a line in some order and each says one statement in sequence: "My number is less than 1," "My number is less than 2," ... "My number is less than 10" (each person makes exactly one statement).
Question: What is the maximum number of truth-tellers that could be among these people? | 8 | 0.375 |
Calculate the mass of silver nitrate:
\[ m\left(\mathrm{AgNO}_{3}\right) = n \cdot M = 0.12 \cdot 170 = 20.4 \text{ g} \]
Determine the mass fraction of potassium sulfate in the initial solution:
\[ \omega\left(\mathrm{AgNO}_{3}\right) = \frac{m\left(\mathrm{AgNO}_{3}\right) \cdot 100\%}{m_{\mathrm{p}-\mathrm{pa}}\left(\mathrm{AgNO}_{3}\right)} = \frac{20.4 \cdot 100\%}{255} = 8\% \] | 8\% | 0.5 |
Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum. | 700000 | 0.375 |
Calculate the angle between the bisectors of the coordinate angles $x O y$ and $y O z$. | 60^\circ | 0.625 |
A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of 30, how many soldiers will be left over? | 23 | 0.875 |
The real polynomial \( p(x) \) is such that for any real polynomial \( q(x) \), we have \( p(q(x)) = q(p(x)) \). Find \( p(x) \). | p(x) = x | 0.875 |
The set of all positive integers can be divided into two disjoint subsets of positive integers \(\{f(1), f(2), \cdots, f(n), \cdots\}, \{g(1), g(2), \cdots, g(n), \cdots\}\), where \(f(1)<f(2)<\cdots<f(n)<\cdots\) and \(g(1)< g(2)<\cdots<g(n)<\cdots\) and it is given that \(g(n)=f[f(n)]+1\) for \(n \geq 1\). Find \(f(240)\). | 388 | 0.25 |
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012? | 2012 | 0.875 |
In a certain country, there are 47 cities. Each city has a bus station from which buses travel to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes originating from each city. It turned out that if we do not consider the city of Ozerny, then for each of the remaining 46 cities, the number of internal routes originating from it differs from the number of routes originating from other cities. Find out how many cities in the country have direct bus connections with the city of Ozerny.
The number of internal bus routes for a given city is the number of cities in the country that can be reached from that city by a direct bus without transfers. Routes are symmetric: if you can travel by bus from city $A$ to city $B$, you can also travel by bus from city $B$ to city $A$. | 23 | 0.75 |
Find the number of three-digit numbers which are multiples of 3 and are formed by the digits 0, 1, 2, 3, 4, 5, 6, and 7 without repetition. | 106 | 0.125 |
In a family of 4 people, if Masha's scholarship is doubled, the total income of the family will increase by 5%. If instead, the mother's salary is doubled, the total income will increase by 15%. If the father's salary is doubled, the total income will increase by 25%. By what percentage will the total income of the family increase if the grandfather's pension is doubled? | 55\% | 0.875 |
Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? | 8654232 | 0.625 |
Pyramid \( E A R L Y \) is placed in \((x, y, z)\) coordinates so that \( E=(10,10,0) \), \( A=(10,-10,0) \), \( R=(-10,-10,0) \), \( L=(-10,10,0) \), and \( Y=(0,0,10) \). Tunnels are drilled through the pyramid in such a way that one can move from \((x, y, z)\) to any of the 9 points \((x, y, z-1)\), \((x \pm 1, y, z-1)\), \((x, y \pm 1, z-1)\), \((x \pm 1, y \pm 1, z-1)\). Sean starts at \( Y \) and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability \(\frac{1}{9}\) each time. What is the probability that he ends up at the point \((8,9,0)\)? | \frac{550}{9^{10}} | 0.125 |
A Christmas garland hanging along the school corridor consists of red and blue bulbs. Next to each red bulb, there is necessarily at least one blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | 33 | 0.25 |
For which values of the parameter \(a\) does the equation \(\frac{\log_{a} x}{\log_{a} 2} + \frac{\log_{x} (2a - x)}{\log_{x} 2} = \frac{1}{\log_{a^2 - 1} 2}\) have (1) a solution? (2) exactly one solution? | 2 | 0.75 |
Find all real numbers \(a\) such that \(a+\frac{2}{3}\) and \(\frac{1}{a}-\frac{3}{4}\) are integers. | \frac{4}{3} | 0.75 |
The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and the number itself) is 10560. Find this number. | 8987 | 0.625 |
Let \( P \) be any interior point of an equilateral triangle \( ABC \). Let \( D, E, \) and \( F \) be the feet of the perpendiculars dropped from \( P \) to the sides \( BC, CA, \) and \( AB \) respectively. Determine the value of the ratio
$$
\frac{PD + PE + PF}{BD + CE + AF}.
$$ | \frac{\sqrt{3}}{3} | 0.375 |
How many natural numbers not exceeding 500 are not divisible by 2, 3, or 5? | 134 | 0.125 |
There are \( n \) vectors in space such that any pair of them forms an obtuse angle. What is the maximum possible value of \( n \)? | 4 | 0.75 |
If Alex does not sing on Saturday, then she has a 70% chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a 50% chance of singing on Sunday, find the probability that she sings on Saturday. | \frac{2}{7} | 0.875 |
Place all terms of the arithmetic sequence \(2, 6, 10, 14, \cdots, 2006\) closely together to form a large number \(A = 261014 \cdots 2006\). Find the remainder when \(A\) is divided by 9. | 8 | 0.625 |
Given the regular hexagon \( A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} \). Show that the diagonals \( A_{1} A_{3}, A_{3} A_{5}, A_{5} A_{1} \) and \( A_{2} A_{4}, A_{4} A_{6}, A_{6} A_{2} \) define a regular hexagon. How does the area of this smaller hexagon compare to the area of the given hexagon? | \frac{1}{3} | 0.75 |
Fill the numbers from 1 to 9 in the spaces of the given diagram (with 2 and 3 already filled), so that the number on the left is greater than the number on the right, and the number above is greater than the number below. How many different ways are there to fill the diagram? | 16 | 0.125 |
What is the limit of the following sequence?
$$
a_{n}=\frac{\sum_{j=0}^{\infty}\binom{n}{2 j} \cdot 2^{j}}{\sum_{j=0}^{\infty}\binom{n}{2 j+1} \cdot 2^{j}}
$$
| \sqrt{2} | 0.875 |
A student did not notice the multiplication sign between two three-digit numbers and wrote a six-digit number, which turned out to be seven times greater than their product. Find these numbers. | 143 | 0.875 |
A farmer bought two sheep in 1996. That year, he didn't have any offspring. The first sheep gave birth to one lamb every three years, and the second sheep gave birth to one lamb every two years. All the sheep born each year also gave birth to one lamb annually. How many sheep will the farmer have in the year 2000? | 9 | 0.375 |
Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the \(n\)th digit from the left with probability \(\frac{n-1}{n}\). (In particular, the 1 is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-3 number. (For example, if the remaining number on the board is 1000, then its value is 27.) | 681751 | 0.125 |
The roots of \( x^3 - x - 1 = 0 \) are \( r, s, t \). Find \( \frac{1 + r}{1 - r} + \frac{1 + s}{1 - s} + \frac{1 + t}{1 - t} \). | -7 | 0.625 |
First-grader Petya was arranging his available chips into the outline of an equilateral triangle such that each side, including the vertices, contains the same number of chips. Then, using the same chips, he managed to arrange them into the outline of a square in the same manner. How many chips does Petya have if the side of the square contains 2 fewer chips than the side of the triangle? | 24 | 0.875 |
Suppose \( n \) is a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2} + y^{2} + z^{2}\right) \leqslant n \left(x^{4} + y^{4} + z^{4}\right)\) always holds. Then the minimum value of \( n \) is \(\qquad\) | 3 | 0.625 |
One day, Papa Smurf conducted an assessment for 45 Smurfs in the Smurf Village. After the assessment, he found that the average score of the top 25 Smurfs was 93 points, and the average score of the bottom 25 Smurfs was 89 points. By how many points does the total score of the top 20 Smurfs exceed that of the bottom 20 Smurfs? | 100 | 0.375 |
If 14! is divisible by \(6^k\), where \(k\) is an integer, find the largest possible value of \(k\). | 5 | 0.625 |
In a movie theater, five friends took seats numbered 1 through 5 (where seat number 1 is the leftmost). During the movie, Anya went to get popcorn. When she returned, she found that Varya had moved three seats to the right, Galya had moved one seat to the left, and Diana and Ella had switched seats, leaving the end seat for Anya. In which seat was Anya sitting before she stood up? | 3 | 0.625 |
In the figure below, $ABCD$ is a trapezoid and its diagonals $AC$ and $BD$ are perpendicular. Additionally, $BC=10$ and $AD=30$.
a) Determine the ratio between the segments $BE$ and $ED$.
b) Find the length of the segments $EC$, $AE$, and $ED$ in terms of the length of $BE=x$.
c) If $AE \cdot EC=108$, determine the value of $BE \cdot ED$. | 192 | 0.125 |
A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm? | 13 \text{ cm} | 0.125 |
If a die is rolled, event \( A = \{1, 2, 3\} \) consists of rolling one of the faces 1, 2, or 3. Similarly, event \( B = \{1, 2, 4\} \) consists of rolling one of the faces 1, 2, or 4.
The die is rolled 10 times. It is known that event \( A \) occurred exactly 6 times.
a) Find the probability that under this condition, event \( B \) did not occur at all.
b) Find the expected value of the random variable \( X \), which represents the number of occurrences of event \( B \). | \frac{16}{3} | 0.75 |
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).
Find the smallest value of \( n \). | 13 | 0.5 |
A point is chosen randomly with uniform distribution in the interior of a circle with a radius of 1. What is its expected distance from the center of the circle? | \frac{2}{3} | 0.75 |
There is a deck of 1024 cards, each of which has a set of different digits from 0 to 9 written on them, and all sets are different (`in particular, there is also an empty card`). We call a set of cards complete if each digit from 0 to 9 appears exactly once on them.
Find all natural numbers \( k \) for which there exists a set of \( k \) cards with the following condition: among them, it is impossible to choose a complete set, but adding any card from the deck would violate this condition. | 512 | 0.25 |
Let the set \( S = \{1, 2, \cdots, 280\} \). Find the smallest positive integer \( n \) such that any \( n \)-element subset \( T \) of \( S \) contains 5 numbers that are pairwise coprime. | 217 | 0.125 |
Every day, Ivan Ivanovich is driven to work by a company car. One day, Ivan Ivanovich decided to walk and left his house an hour earlier than usual. On the way, he met the company car, and he completed the remaining distance in the car. As a result, he arrived to work 10 minutes earlier than usual. How long did Ivan Ivanovich walk? | 55 \text{ minutes} | 0.25 |
An electronic watch shows 6:20:25 at 6 hours, 20 minutes, and 25 seconds. Within the one-hour period from 5:00:00 to 5:59:59, how many instances are there where all five digits of the time displayed are different? | 840 | 0.625 |
There are two cylinders with a volume ratio of 5:8. Their lateral surfaces can be unfolded into identical rectangles. If the length and width of this rectangle are both increased by 6, the area increases by 114. What is the area of this rectangle? | 40 | 0.25 |
Find the smallest natural number that can be represented in the form $13x + 73y$ in three different ways, where $x$ and $y$ are natural numbers. | 1984 | 0.375 |
A polynomial of degree 3n has the value 2 at 0, 3, 6, ... , 3n, the value 1 at 1, 4, 7, ... , 3n-2, and the value 0 at 2, 5, 8, ..., 3n-1. Its value at 3n+1 is 730. What is n? | 4 | 0.125 |
Given an equilateral triangle \( \triangle ABC \) with each side having a length of 2. If \(\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC})\) and \(\overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2} \overrightarrow{BC}\), find the area of \(\triangle APQ\). | \frac{\sqrt{3}}{3} | 0.5 |
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