meetforever/GRPO_CL huggingface dataset:
user: In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.
Answer: 23
user: Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$.
Answer: 5
user: The difference between the maximal and the minimal diagonals of the regular $n$-gon equals its side ($n > 5$). Find $n$.
Answer: 9
user: A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number of passengers that the tram could have?
Answer: 20
user: Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$, and $a+b$ are all distinct squares. What is the smallest possible value of $b$?
Answer: 21
user: Determine the value of the parameter $m$ such that the equation $(m-2)x^2 + (m^2-4m+3)x - (6m^2-2) = 0$ has real solutions, and the sum of the cubes of these solutions is equal to zero.
Answer: 3
user: Find all positive integers $l$ for which the equation $$a^3+b^3+ab=(lab+1)(a+b)$$ has a solution over positive integers $a,b$.
Answer: 1
user: Given a regular polygon with $n$ sides, it is known that there are 1200 ways to choose three of the vertices of the polygon such that they form the vertices of a right triangle. What is the value of $n$?
Answer: 50
user: Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:

1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or  
2. From $P$ and $Q$ to $5P-4Q$ and $5Q-4P$.

The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.

At the beginning of the game, the two numbers on the board are $2024$ and $A$. It is known that Kobar does not lose on his first move. Determine the largest possible value of $A$ so that Borah can win this game.
Answer: 2179
user: In quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $
abla ABD = 36^{\circ}$ and $
abla BCD = 54^{\circ}$, find the measure of angle $
abla ADC$ in degrees.
Answer: 102
user: Find all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ and $1 + n = 5(d_{1} + d_{2} + d_{3} + d_{4})$.
Answer: 1519
user: Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square.
Answer: 25
user: For some real number $c$, the graphs of the equation $y = |x - 20| + |x + 18|$ and the line $y = x + c$ intersect at exactly one point. What is $c$?
Answer: 18
user: Find all positive integers $n$ such that $n^3 - 5n^2 + 9n - 6$ is a perfect square number.
Answer: 2
user: Find all values of the parameter $a$ for which the sum of all real solutions of the equation $x^4 - 5x + a = 0$ is equal to $a$.
Answer: 0
user: Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares?
Answer: 1
user: Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-complete if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n$ divides $d_1$ and gcd$(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there?
Answer: 32
user: Let $p$ be a polynomial with integer coefficients such that $p(15) = 6$, $p(22) = 1196$, and $p(35) = 26$. Find an integer $n$ such that $p(n) = n + 82$.
Answer: 28
user: Find the smallest positive integer $N$ satisfying the following three properties:
- $N$ leaves a remainder of $5$ when divided by $7$.
- $N$ leaves a remainder of $6$ when divided by $8$.
- $N$ leaves a remainder of $7$ when divided by $9$.
Answer: 502
user: A time is chosen randomly and uniformly in a 24-hour day. The probability that at that time, the non-reflex angle between the hour hand and minute hand on a clock is less than $\frac{360}{11}$ degrees is $\frac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m + n$.
Answer: 211
user: Find all integers $n$ such that $1 < n < 10^6$ and $n^3 - 1$ is divisible by $10^6 n - 1$.
Answer: 1000
user: Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every subsequent term is obtained by multiplying its preceding term by $a$ and adding $b$. 

For example, if $a = 2$ and $b = 4$, the first three terms are $5, 14, 32$. 

What is the maximum possible number of prime numbers that can occur before encountering the first composite term?
Answer: 5
user: Find the sum of all integers $n$ such that $n^4 + n^3 + n^2 + n + 1$ is a perfect square.
Answer: 2
user: Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\frac{m}{n}$ for relatively prime positive integers $m$, $n$. Compute $100m + n$.
Answer: 1116
user: Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression:

$$ \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2} $$

If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $\text{GCD}(m,n)=1$, find the value of $m+n$.
Answer: 25
user: Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of this row. Assume that each box must be placed so that it straddles two lower boxes. Including the arrangement with no boxes, how many arrangements can Mereduth make?
Answer: 21
user: We are given weights ranging from 1 to 5771, i.e., 1, 2, 3, ..., 5770, 5771. These weights are to be partitioned into $n$ sets such that each set has an equal total weight. Determine the maximal value of $n$ for which this partitioning is possible.
Answer: 2886
user: Determine all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square, i.e., the square of an integer.
Answer: 5
user: Find the smallest positive integer $n$ such that $n$ is divisible by $100$ and has exactly $100$ divisors.
Answer: 162000
user: A frog is located at the origin of the Cartesian coordinate system. Every second, the frog jumps either horizontally or vertically to one of the four adjacent points with integer coordinates. Determine the number of different points where the frog can be located after 2015 seconds.
Answer: 4064256
user: Numbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax^2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.
Answer: 9
user: There are $64$ booths arranged around a circular table, each containing a chip. The chips and booths are numbered from $1$ to $64$. At the center of the table, there are $1996$ light bulbs, all initially turned off. 

Every minute, the chips move simultaneously in a circular pattern as follows: chip $1$ moves one booth forward, chip $2$ moves two booths forward, and so on, such that multiple chips can occupy the same booth at the same time. 

A bulb is lit whenever chip $1$ shares a booth with any other chip. 

Determine the position of chip $1$ on the first minute when all $1996$ bulbs are lit.
Answer: 32
user: Find the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
Answer: 8
user: Consider an isosceles triangle $ABC$ with sides $BC = 30$, $CA = AB = 20$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $M$ be the midpoint of $AD$. Let $PQ$ be a chord of the circumcircle of triangle $ABC$, such that $M$ lies on $PQ$ and $PQ$ is parallel to $BC$. Determine the length of $PQ$.
Answer: 25
user: Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:

1. In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.

2. In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line.

3. In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.

Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
Answer: 0
user: There are $12$ students in a classroom; $6$ of them are Democrats and $6$ of them are Republicans. Every hour, the students are randomly separated into four groups of three for political debates. If a group contains students from both parties, the minority in the group will change their political alignment to that of the majority at the end of the debate. What is the expected amount of time needed for all $12$ students to have the same political alignment, in hours?
Answer: 6
user: $ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$ and $E$ on sides $AB$ and $AC$ respectively such that $AD = DE = EC = n$, where $n$ is also an integer. Find the value of $m$.
Answer: 30
user: Find all prime numbers $p$ such that the number of distinct positive factors of $p^2 + 2543$ is less than 16.
Answer: 2
user: Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$.
Answer: 24
user: Let $D$, $E$, and $F$ respectively be the feet of the altitudes from $A$, $B$, and $C$ of acute triangle $\triangle ABC$ such that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.
Answer: 14
user: Find all positive integers $n$ such that there exists a prime number $p$ where $p^n - (p-1)^n$ is a power of $3$.

Note: A power of $3$ is a number of the form $3^a$, where $a$ is a positive integer.
Answer: 2
user: Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
Answer: 86
user: In the tribe of Zimmer, a boy is not considered a man until he completes a rite of passage involving a series of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each subsequent hike is along a different road and is $\frac{1}{4}$ kilometer longer than the previous hike. The rite of passage is completed when the cumulative distance walked exceeds $1000$ kilometers. How many roads must a man walk down before he is considered a man in the tribe of Zimmer?
Answer: 73
user: Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$

Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
Answer: 2047
user: Determine the smallest positive integer $A$ with an odd number of digits such that both $A$ and the number $B$, created by removing the middle digit of $A$, are divisible by $2018$.
Answer: 100902018
user: Determine all positive integers $n$ for which the equation
\[
x^n + (2+x)^n + (2-x)^n = 0
\]
has an integer as a solution.
Answer: 1
user: Matilda drew $12$ quadrilaterals. The first quadrilateral is a rectangle with integer sides, where the width is $7$ times the length. Every time she drew a quadrilateral, she joined the midpoints of each pair of consecutive sides with a segment. It is known that the last quadrilateral Matilda drew was the first with an area less than $1$. What is the maximum area possible for the first quadrilateral?

\[ \text{Note: The above figure illustrates the first two quadrilaterals that Matilda drew.} \]
Answer: 2023
user: Determine all real solutions of the equation:
\[
\frac{x^{2}}{x-1} + \sqrt{x-1} + \frac{\sqrt{x-1}}{x^{2}} = \frac{x-1}{x^{2}} + \frac{1}{\sqrt{x-1}} + \frac{x^{2}}{\sqrt{x-1}}.
\]
Answer: 2
user: Find the number of values of $x$ such that the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
Answer: 4224
user: Determine all functions $f : \mathbb{N}_0 \to \mathbb{R}$ satisfying $f(x+y) + f(x-y) = f(3x)$ for all $x, y$.
Answer: 0
user: Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum.
Answer: 3
user: Five airway companies operate in a country consisting of 36 cities. Between any pair of cities, exactly one company operates two-way flights. If some air company operates between cities $A, B$ and $B, C$, we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged, there are at least $k$ properly-connected triples.
Answer: 3780
user: In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. What is the smallest amount of rubles you can spend to rearrange the chips in reverse order?
Answer: 50
user: Find all positive integers $n$ such that all positive divisors of $n$ can be arranged in a rectangular table with the following constraints:

- Each cell contains a distinct divisor.
- The sums of all rows are equal.
- The sums of all columns are equal.
Answer: 1
user: In the game of *Winners Make Zeros*, a pair of positive integers \((m,n)\) is written on a sheet of paper. Then the game begins, as the players make the following legal moves:

- If \(m \geq n\), the player chooses a positive integer \(c\) such that \(m-cn \geq 0\), and replaces \((m,n)\) with \((m-cn,n)\).
- If \(m < n\), the player chooses a positive integer \(c\) such that \(n-cm \geq 0\), and replaces \((m,n)\) with \((m,n-cm)\).

When \(m\) or \(n\) becomes \(0\), the game ends, and the last player to have moved is declared the winner. If \(m\) and \(n\) are originally \(2007777\) and \(2007\), find the largest choice the first player can make for \(c\) (on his first move) such that the first player has a winning strategy after that first move.
Answer: 999
user: Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.
Answer: 3825
user: Find all natural numbers $n > 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$.
Answer: 1342
user: One hundred and one of the squares of an $n \times n$ table are colored blue. It is known that there exists a unique way to cut the table into rectangles along the boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.
Answer: 101
user: A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way that each newly drawn diagonal intersects no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
Answer: 4016
user: Find the sum of all positive integers $n$ such that $\frac{2n+1}{n(n-1)}$ has a terminating decimal representation.
Answer: 52
user: For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized?
Answer: 11
user: Bob starts with an empty whiteboard. He then repeatedly chooses one of the digits $1, 2, \ldots, 9$ (uniformly at random) and appends it to the end of the currently written number. Bob stops when the number on the board is a multiple of $25$. Let $E$ be the expected number of digits that Bob writes. If $E = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$.
Answer: 8102
user: Hari is obsessed with cubics. He comes up with a cubic polynomial with leading coefficient 1, rational coefficients, and real roots $0 < a < b < c < 1$. He knows the following three facts:  
1. $P(0) = -\frac{1}{8}$
2. The roots form a geometric progression in the order $a, b, c$.
3. \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \frac{9}{2} \]

The value $a + b + c$ can be expressed as $\frac{m}{n}$, where $m, n$ are relatively prime positive integers. Find $m + n$.
Answer: 31
user: $A$ and $B$ are on a circle with radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen such that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?
Answer: 20
user: Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4, \ldots, n\}$ into two sets, at least one of these sets contains three (not necessarily distinct) numbers $a, b, c$ for which $ab = c$.
Answer: 243
user: What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?
Answer: 35
user: Find all prime numbers $p$ such that both $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.
Answer: 5
user: Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 7
user: Define $A(n)$ as the average of all positive divisors of the positive integer $n$. Find the sum of all solutions to $A(n) = 42$.
Answer: 1374
user: $f(x), g(x)$ are two quadratic trinomials and $a, b, c, d$ are some real numbers. Given the following values:
- $f(a) = 2$
- $f(b) = 3$
- $f(c) = 7$
- $f(d) = 10$
- $g(a) = 16$
- $g(b) = 15$
- $g(c) = 11$

Find the value of $g(d)$.
Answer: 8
user: Find all prime numbers $p$ such that the expression $p^2 - p + 1$ is a perfect cube.
Answer: 19
user: There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5 + 2004z = 1$, then $P(z^2) = 0$. Calculate the quotient $\frac{P(1)}{P(-1)}$.
Answer: 1
user: Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1, m_2, n_1, n_2$ such that $a = m_1^2 + n_1^2$, $a^2 = m_2^2 + n_2^2$, and $m_1 - n_1 = m_2 - n_2$.
Answer: 261
user: Find all positive integers $n$ such that the number $\frac{(2n)!+1}{n!+1}$ is a positive integer.
Answer: 3
user: On a $1000 \times 1000$-board, we place dominoes such that each domino covers exactly two squares on the board. Additionally, two dominoes are not allowed to be adjacent but are allowed to touch at a vertex. Determine the maximum number of dominoes that can be placed on the board in this way.
Answer: 250000
user: Find all positive integers $r$ such that there exist positive prime numbers $p$ and $q$ satisfying the equation $$p^2 + pq + q^2 = r^2.$$
Answer: 7
user: Triangle $ABC$ has side lengths $AB = 20$, $AC = 14$, and $BC = 22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\frac{p}{q} = \frac{[AMPN]}{[ABC]}$ for positive integers $p$ and $q$ that are coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p + q$.
Answer: 331
user: A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$.
Answer: 2842
user: Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a + b + c = d$.
Answer: 11
user: Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$, and $b+2$. He then writes all six pairwise products of the numbers in the notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.
Answer: 2
user: Determine the real number $a$ such that $f(a) = a$ is a relative minimum of the function $f(x) = x^4 - x^3 - x^2 + ax + 1$.
Answer: 1
user: Let $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line $AP$, point $B$ determines point $Q$, as seen in the figure. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$.
Answer: 20
user: Solve the equation:
\[
  \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for \(x\) where \(0 < x < 90\).
Answer: 9
user: There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?
Answer: 20
user: Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice, where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many people are in Cambridge?
Answer: 7
user: Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a + b + c = 2014$, find $c$.
Answer: 1440
user: Suppose an integer $x$, a natural number $n$, and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Find the largest value of $p$.
Answer: 47
user: Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
Answer: 9
user: A triangle has side lengths of $x$, $75$, and $100$, where $x < 75$, and altitudes of lengths $y$, $28$, and $60$, where $y < 28$. What is the value of $x + y$?
Answer: 56
user: A square has been divided into $2022$ rectangles, with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles?
Answer: 2025
user: In the diagram, $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^\circ$ and $\angle BOE = 68^\circ$. Find $\angle AOF$.
Answer: 118
user: Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy and hops till he falls asleep with $0$ energy. How many different places could he have gone to sleep?
Answer: 10201