ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In triangle $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are pairwise relatively prime integers, find $a + b + c$.
Answer: 34
Problem: Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Answer: 113
Problem: Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.
Answer: -3
Problem: There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in the other direction, placing the eighth card in box $6$, the ninth in box $5$, up to the thirteenth card being placed in box $1$. The fourteenth card is then placed in box $2$, and this continues until every card is distributed. What box will the last card be placed in?
Answer: 3
Problem: Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 37
Problem: For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?
Answer: 10
Problem: Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is half of the arithmetic mean of the numbers $a, b, c, d$.
Answer: 10
Problem: $n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?
Answer: 2
Problem: Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.
Answer: 509
Problem: Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.
Answer: 3
Problem: There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.
Answer: 1007
Problem: Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.
Answer: 28
Problem: Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.
Answer: 5
Problem: Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight), a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?
Answer: 5
Problem: Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
Answer: 1
Problem: Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of each fruit and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?
Answer: 15
Problem: Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
Answer: 10
Problem: Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$
Answer: 402
Problem: For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ for which $S(n) - S(n + a) = 2018$.
Answer: 7
Problem: Find all integers $k$ such that all roots of the following polynomial are also integers:

$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$
Answer: 5
Problem: A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer candies, possibly receiving none. Determine the smallest value of $N$ for which this distribution is possible, regardless of the number of tasks in the Olympiad and the students' successes.
Answer: 300
Problem: If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.
Answer: 1
Problem: Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 37
Problem: Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves:  
1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$.  
2. If there are 2 numbers (written on the board) in the form $n, n+4$, we can erase them and write down $n-1$.  

After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that independently of the starting numbers, each number which appears in any move is greater than or equal to $c$.
Answer: -3
Problem: Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.
Answer: 3
Problem: Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integers. He first rolls the die, and if the number showing on top is $k$, he then uniformly and randomly selects a $k$-element subset from his list and calculates the sum of the numbers in his subset. Given that the expected value of this sum is $2020$, compute the sum of all possible values of $n$.
Answer: 8556
Problem: The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.

What is the smallest value of $n$ for which the second player can ensure that he completes both his moves?
Answer: 14
Problem: Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
Answer: 7
Problem: The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.
Answer: 3
Problem: Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.
Answer: 30
Problem: Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one to take the last stone from the pile. Assuming Dexter and Raquel play optimally, compute the number of positive integers $N \le 2021$ where Dexter wins this game.
Answer: 1010
Problem: Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$.
Answer: 40
Problem: Find the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.
Answer: 9
Problem: Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.
Answer: 10
Problem: Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x) = a$. The game continues until Petya is forced to repeat an answer. What is the minimal amount of rubles Vasya must pay in order to win?
Answer: 4
Problem: Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.
Answer: 64
Problem: On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum of the erased numbers divided by 13 is written on the board. After a few repetitions of this process, only three numbers are left, of which two are 9 and 999. Find the third number.
Answer: 8
Problem: For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\frac{s(n)}{n}$?
Answer: 1099
Problem: Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.
Answer: 21
Problem: How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?

(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
Answer: 40320
Problem: Find a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$ has a rational root.
Answer: 1
Problem: Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.
Answer: 5
Problem: Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.
Answer: 0
Problem: Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls, and 463 blue balls, or one can choose 360 red balls and 540 blue balls.
Answer: 270901
Problem: Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y$.
Answer: 12
Problem: Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.
Answer: 31
Problem: For any real numbers $x$ and $y$ that satisfy the equations:
\[
x + y - xy = 155
\]
and
\[
x^2 + y^2 = 325
\]
Find $|x^3 - y^3|$.
Answer: 4375
Problem: There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.
Answer: 45
Problem: Find all positive integers $k$ for which the equation:
$$ \text{lcm}(m,n) - \text{gcd}(m,n) = k(m-n) $$
has no solution in positive integers $(m,n)$ with $m \neq n$.
Answer: 2
Problem: Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.
Answer: 1
Problem: Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
- Those that contain the faces of the tetrahedron, and
- Each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.

Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.
Answer: 3
Problem: For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.
Answer: 2021
Problem: Suppose integers $a < b < c$ satisfy the following conditions:
\[
 a + b + c = 95 \\
 a^2 + b^2 + c^2 = 3083
\]
Find the value of $c$.
Answer: 37
Problem: Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.
Answer: 18
Problem: Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct.
Answer: 132
Problem: Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.

Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
Answer: 1003
Problem: Byan is playing a game called "raven, raven, falcon" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either "raven" or "falcon," each with a probability of $\frac{1}{2}$. The game ends when Byan has said "falcon" twice. The probability that one of his friends will be called a "falcon" twice can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
Answer: 107
Problem: Find all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular table under 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
Problem: One day, students in school organized an exchange among themselves with the following rates:
- $11$ strawberries exchange for $14$ raspberries.
- $22$ cherries exchange for $21$ raspberries.
- $10$ cherries exchange for $3$ bananas.
- $5$ pears exchange for $2$ bananas.

The question is: How many pears does Amila need to give to get $7$ strawberries?
Answer: 7
Problem: In triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.
Answer: 80
Problem: Find all integers $n$ such that $\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.
Answer: 0
Problem: Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.
Answer: 10
Problem: The numbers $1, 2, \ldots, 10000$ are placed in the cells of a $100 \times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the numbers differ by exactly $5000$. Let $S$ be the minimum of these distances. What is the largest value $S$ can take?
Answer: 50
Problem: The triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same point $E$. Determine the length of $CD$.
Answer: 152
Problem: An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as the third.
Answer: 45
Problem: Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, including $1$ and $n$.
Answer: 9
Problem: Determine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural number.
Answer: 1
Problem: Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.
Answer: 200
Problem: It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.
Answer: 1987
Problem: Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$.
Answer: 3
Problem: Find all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numbers $a^3 - 1$, $a^2 - 1$.
Answer: 2
Problem: Given that nonzero real numbers $x$ and $y$ satisfy the equations $x + \frac{1}{y} = 3$ and $y + \frac{1}{x} = 4$, find the value of $xy + \frac{1}{xy}$.
Answer: 10
Problem: Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.
Answer: 89
Problem: Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.
Answer: 1
Problem: Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 digits all equal to 9.
Answer: 34
Problem: Let the number of ways for a rook to return to its original square on a $4 \times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.
Answer: 4
Problem: \( \triangle ABC \) is an isosceles triangle with \( AB = AC \). Point \( P \) is located inside \( \triangle ABC \) such that \( \angle BCP = 30^\circ \), \( \angle APB = 150^\circ \), and \( \angle CAP = 39^\circ \). Find \( \angle BAP \).
Answer: 13
Problem: Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.
Answer: 720
Problem: The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?

\[
\textbf{a)}\ 0 \quad \textbf{b)}\ 2 \quad \textbf{c)}\ 4 \quad \textbf{d)}\ 6 \quad \textbf{e)}\ 8
\]
Answer: 2
Problem: Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:

1. $f(n)$ is a positive integer.
2. $f(n + 1) > f(n)$.
3. $f(f(n)) = 3n$.

Find $f(2001)$.
Answer: 3816
Problem: Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.
Answer: 40
Problem: Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$.
Answer: -1
Problem: Find all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.
Answer: 5
Problem: Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$.
Answer: 7
Problem: Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:
\[
\begin{align*}
x(y^2+yz+z^2)&=3y+10z,\\
y(z^2+zx+x^2)&=21z+24x,\\
z(x^2+xy+y^2)&=7x+28y.
\end{align*}
\]
Find $xy+yz+zx$.
Answer: 31