meetforever/GRPO_CL huggingface dataset:
user: Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Answer: 415
user: Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and $DC = 16$, determine $AC^2$.
Answer: 936
user: Petya gave Vasya a number puzzle. Petya chose a digit $X$ and said, "I am thinking of a three-digit number that is divisible by 11. The hundreds digit is $X$ and the tens digit is 3. Find the units digit." Vasya was excited because he knew how to solve this problem, but then realized that the problem Petya gave did not have an answer. What digit $X$ did Petya choose?
Answer: 4
user: A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.

Ada has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.

Ada has written forty haikus so far and writes down seven haikus every hour. At which hour after she begins will she not have enough haikus done?
Answer: 4
user: The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimeter of polygon \( X \), then the equation \( P(APQRD) = P(ABCD) + 32 \) holds. Determine the length of side \( CD \).
Answer: 10
user: We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1 \times 1$ squares with an area of $2010 \times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction, find $a + b$.
Answer: 6029
user: Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ at points $A$ and $B$. The intersection of tangents to $G_2$ at points $A$ and $B$ is point $C$. If $C$ lies on $G_1$, find all possible values of $p$.
Answer: 2
user: Let $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:
- $A'B' \parallel AB$, 
- $C'C$ is the bisector of angle $A'C'B'$, 
- $A'C' + B'C' = AB$.
Answer: 1
user: The radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, with $p, q$ being prime numbers and $m, n$ being positive integers. Determine $r$.
Answer: 5
user: Using each of the digits $1, 2, 3, \ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Their sum ends in $n$ zeroes, where $n$ is a non-negative integer. Determine the maximum possible value of $n$.
Answer: 8
user: Determine the largest positive integer $n$ such that the following statement holds:

If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.
Answer: 107
user: Solve for $x$:
\[
v - w + x - y + z = 79 \\
v + w + x + y + z = -1 \\
v + 2w + 4x + 8y + 16z = -2 \\
v + 3w + 9x + 27y + 81z = -1 \\
v + 5w + 25x + 125y + 625z = 79.
\]
Answer: 24
user: You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:

1. Add 1 to the unit's digit.
2. Subtract 1 from the ten's digit.
3. Add 1 to the hundred's digit.
4. Subtract 1 from the thousand's digit.

If the resulting number is also a perfect square, find the original number. Is it unique?
Answer: 3025
user: Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron?
Answer: 864
user: Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:

\[
\begin{align*}
700 \cdot 712 \cdot 718 + 320,\\
701 \cdot 713 \cdot 719 + 320,\\
702 \cdot 714 \cdot 720 + 320,\\
&\vdots\\
798 \cdot 810 \cdot 816 + 320,\\
799 \cdot 811 \cdot 817 + 320.
\end{align*}
\]

She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
Answer: 7
user: Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.
Answer: 18
user: The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of $a + b + c$.
Answer: 31
user: In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points equal to the absolute difference of their numbers. Determine the maximum number of points that can be guaranteed after $2014$ steps.
Answer: 1007
user: Find all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square.
Answer: 3
user: In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offers direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city on the island "Mayor" to any other city on the island "Mayor".

If the percentage of flights that connect two cities in the same state is the same for the flights of each airline, what is the least number of cities that can be on the island "Mayor"?
Answer: 65
user: Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?
Answer: 100
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
user: In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. Similarly, some number of the particles from the right side of the tube move to the left at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
Answer: 102
user: 30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maximum number of **astonishing** teams.
Answer: 1
user: In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following options:

\[ \textbf{(A)}\ 288 \quad \textbf{(B)}\ 291 \quad \textbf{(C)}\ 294 \quad \textbf{(D)}\ 297 \quad \textbf{(E)}\ 300 \]
Answer: 300
user: Let $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.
Answer: 48
user: For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.
Answer: 6
user: Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\[
x_1(x_1+1) = A \\
x_2(x_2+1) = A \\
x_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.
\]
Answer: -7
user: Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$.
Answer: 45
user: A function $f$ is defined on integers such that:

- $f(n) = n + 3$ if $n$ is odd.
- $f(n) = \frac{n}{2}$ if $n$ is even.

If $k$ is an odd integer, determine the values for which $f(f(f(k))) = k$.
Answer: 1
user: Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.
Answer: 1594
user: 10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\frac{1}{2}$ probability). 

Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.
Answer: 828
user: On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$.
Answer: 8
user: Suppose in the plane, there are 10 pairwise nonparallel lines intersecting one another. What is the maximum possible number of polygons (with finite areas) that can be formed?
Answer: 36
user: The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?
Answer: 108
user: Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.
Answer: 76
user: A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:

- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$.
- It is divisible by $11$.

What is the sum of all possible numbers of cows that the herder has?

$\textbf{(A) }343 \ \textbf{(B) }494 \ \textbf{(C) }615 \ \textbf{(D) }635 \ \textbf{(E) }726$
Answer: 726
user: All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?
Answer: 6
user: Let $f(n)$ be a function that fulfills the following properties:

- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.

Find $\displaystyle f(2009^{2009^{2009}})$ and justify your answer.
Answer: 0
user: Find all integers $n > 1$ such that every prime that divides $n^6 - 1$ also divides $n^5 - n^3 - n^2 + 1$.
Answer: 2
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:
- 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.
- 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.
- 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: Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.
Answer: 2
user: Find all positive integers $n$ such that $n = 100 \times d(n)$, where $d(n)$ represents the number of positive divisors of $n$.
Answer: 2000
user: In $\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?
Answer: 3
user: Find the number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$.
Answer: 4
user: Solve in the real numbers the equation $3^{x+1} = (x-1)(x-3)$.
Answer: 0
user: Let $f(x) = x^3 + x + 1$. Suppose $g$ is a cubic polynomial such that $g(0) = -1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$.
Answer: 899
user: A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?
Answer: 2019
user: Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
Answer: 32
user: Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have the following table:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
p(n) & 1 & 2 & 5 & 15 & 52 & 203 & 877 & 4140 & 21147 & 115975 \\
\hline
\end{array}
\]

Define a "splitting" of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?
Answer: 17007
user: In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.
Answer: 7
user: Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\frac{n^3 + m}{n + 2}$ is an integer.
Answer: 8
user: What is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for which the rightmost three digits of $2013^n$ are $001$.
Answer: 100
user: A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
Answer: 8
user: Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k \geq 2$ such that $3^n = x^k + y^k$.
Answer: 2
user: Circle $B$, which has radius 2008, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius 1 and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. Circle $C_2$ has radius larger than 1 and is tangent to line $A$ and both circles $B$ and $C_1$. For $n > 1$, circle $C_n$ is tangent to line $A$ and both circles $B$ and $C_{n-1}$. Find the largest value of $n$ such that this sequence of circles can be constructed through circle $C_n$ where the $n$ circles are all tangent to line $A$ at points to the right of $P$.
Answer: 45
user: There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) From each town, there are exactly $k$ direct routes to $k$ other towns.
2) If two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
Answer: 6
user: Let $x, y$ be positive integers such that:
\[
x^4 = (x-1)(y^3 - 23) - 1
\]
Find the maximum possible value of $x + y$.
Answer: 7
user: If the four sides of a quadrilateral are $2$, $3$, $6$, and $x$, find the sum of all possible integral values for $x$.
Answer: 54
user: Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\frac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\frac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)?
Answer: 2500
user: Blue rolls a fair $n$-sided die numbered with integers from $1$ to $n$, and then flips a coin. The coin is weighted to land heads either $\frac{1}{3}$ or $\frac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\frac{1}{15}$, find $n$.
Answer: 10
user: A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?
Answer: 2
user: Let $m,n$ be natural numbers such that $m + 3n - 5 = 2\text{LCM}(m,n) - 11\text{GCD}(m,n)$. Find the maximum possible value of $m+n$.
Answer: 70
user: 25 checkers are placed on the 25 leftmost squares of a $1 \times N$ board. A checker can either move to the empty adjacent square to its right or jump over an adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find the minimal $N$ such that all the checkers can be placed in a row of 25 successive squares but in the reverse order.
Answer: 50
user: Suppose $1$, $2$, and $3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.
Answer: 36
user: Find all positive integers $n$ for which $2^n + 2021n$ is a perfect square.
Answer: 4
user: How many different ways are there to write 2004 as a sum of one or more positive integers which are all "approximately equal" to each other? Two numbers are called approximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.
Answer: 2004
user: Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.
Answer: 5
user: A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. Determine, with proof, the maximum number of distinct license plates that the state can use.
Answer: 100000
user: In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.
Answer: 2000
user: Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100q + p$ is a perfect square.
Answer: 179
user: It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.
Answer: 5
user: Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.
Answer: 2011
user: Determine the value of $k$ for the quartic equation:

$$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$

Given that the product of two of its four roots is $-32$.
Answer: 86
user: There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
Answer: 50
user: There is a pile of eggs. Joan counted the eggs, but her count was off by $1$ in the $1$'s place. Tom counted the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the correct count. When these seven people added their counts together, the sum was $3162$. How many eggs were in the pile?
Answer: 439
user: Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
Answer: 4
user: Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$.
Answer: 2
user: A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n + 76$ equal squares. Find the value of $n$.
Answer: 324