ability stringclasses 1
value | reward_model dict | problem stringlengths 259 3.43k | answer stringlengths 1 23 | source stringclasses 1
value |
|---|---|---|---|---|
MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 = ... | 34 | math_dapo |
MATH | {
"ground_truth": "113",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 t... | 113 | math_dapo |
MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | -3 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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.... | 3 | math_dapo |
MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 37 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?
Remember to put your answer on its own line after "Answer:". | 10 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 arithmet... | 10 | math_dapo |
MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ar... | 2 | math_dapo |
MATH | {
"ground_truth": "509",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.
Remember to put your answer on its own ... | 509 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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,... | 3 | math_dapo |
MATH | {
"ground_truth": "1007",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $2015$ trav... | 1007 | math_dapo |
MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, a... | 28 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $(a-2)^3+(b-2)^3+(c-2)^3=-18$, fin... | 5 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 5 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 bla... | 1 | math_dapo |
MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 15 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 b... | 10 | math_dapo |
MATH | {
"ground_truth": "402",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$
Remember to put ... | 402 | math_dapo |
MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 inf... | 7 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find all integers $k$ such that all roots of the following polynomial are also integers:
$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$
Remember ... | 5 | math_dapo |
MATH | {
"ground_truth": "300",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performa... | 300 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $a + b$.
Remember to put your answer ... | 1 | math_dapo |
MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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)\i... | 37 | math_dapo |
MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | -3 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.
Remember to... | 3 | math_dapo |
MATH | {
"ground_truth": "8556",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 posit... | 8556 | math_dapo |
MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 card... | 14 | math_dapo |
MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $... | 7 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 th... | 3 | math_dapo |
MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a... | 30 | math_dapo |
MATH | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 t... | 1010 | math_dapo |
MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectivel... | 40 | math_dapo |
MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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.
Reme... | 9 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 10 | math_dapo |
MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 a... | 4 | math_dapo |
MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.
Remember to put your answer on its own line after "Ans... | 64 | math_dapo |
MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 nu... | 8 | math_dapo |
MATH | {
"ground_truth": "1099",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 1099 | math_dapo |
MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 hav... | 21 | math_dapo |
MATH | {
"ground_truth": "40320",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 col... | 40320 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 1 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 5 | math_dapo |
MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.
Remember to put your answer on its own... | 0 | math_dapo |
MATH | {
"ground_truth": "270901",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 270901 | math_dapo |
MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 t... | 12 | math_dapo |
MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 31 | math_dapo |
MATH | {
"ground_truth": "4375",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
For any real numbers $x$ and $y$ that satisfy the equations:
\[
x + y - xy = 155
\]
and
\[
x^2 + y^2 = 325
\]
Find $|x^3 - y^3|$.
Remember to... | 4375 | math_dapo |
MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 suc... | 45 | math_dapo |
MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $... | 2 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 1 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $... | 3 | math_dapo |
MATH | {
"ground_truth": "2021",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 2021 | math_dapo |
MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $c$.
Rememb... | 37 | math_dapo |
MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $a^4 + b^4 + c^4$.
Remember t... | 18 | math_dapo |
MATH | {
"ground_truth": "132",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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.
Remem... | 132 | math_dapo |
MATH | {
"ground_truth": "1003",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 o... | 1003 | math_dapo |
MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 107 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 1 | math_dapo |
MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
One day, students in school organized an exchange among themselves with the following rates:
- $11$ strawberries exchange for $14$ raspberries... | 7 | math_dapo |
MATH | {
"ground_truth": "80",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 80 | math_dapo |
MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find all integers $n$ such that $\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.
Remember to put your answer on its own line... | 0 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.
Remember to put your answer on its o... | 10 | math_dapo |
MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 arran... | 50 | math_dapo |
MATH | {
"ground_truth": "152",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $ABD$ ... | 152 | math_dapo |
MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 num... | 45 | math_dapo |
MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 num... | 9 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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.
Remember to put ... | 1 | math_dapo |
MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $... | 200 | math_dapo |
MATH | {
"ground_truth": "1987",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 ... | 1987 | math_dapo |
MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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)... | 3 | math_dapo |
MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $a^3 - 1$, $... | 2 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 + \fr... | 10 | math_dapo |
MATH | {
"ground_truth": "89",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.
Remember to ... | 89 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 t... | 1 | math_dapo |
MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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.
R... | 34 | math_dapo |
MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $k$. Fi... | 4 | math_dapo |
MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
\( \triangle ABC \) is an isosceles triangle with \( AB = AC \). Point \( P \) is located inside \( \triangle ABC \) such that \( \angle BCP =... | 13 | math_dapo |
MATH | {
"ground_truth": "720",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 720 | math_dapo |
MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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... | 2 | math_dapo |
MATH | {
"ground_truth": "3816",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 + ... | 3816 | math_dapo |
MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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 $E$... | 40 | math_dapo |
MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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{... | -1 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.
Remember to put your answer on its own line after "Ans... | 5 | math_dapo |
MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$.
Remember to put your answer on it... | 7 | math_dapo |
MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the 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)&=21... | 31 | math_dapo |
MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Given a pentagon $ABCDE$ with the following properties:
1. Angles $\angle A$ and $\angle C$ are right angles.
2. The side lengths are $|AB| =... | 90 | math_dapo |
MATH | {
"ground_truth": "1440",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.
Remember to put your answer... | 1440 | math_dapo |
MATH | {
"ground_truth": "415",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 tang... | 415 | math_dapo |
MATH | {
"ground_truth": "936",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ re... | 936 | math_dapo |
MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 hundre... | 4 | math_dapo |
MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 ... | 4 | math_dapo |
MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( B... | 10 | math_dapo |
MATH | {
"ground_truth": "6029",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 ... | 6029 | math_dapo |
MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ ... | 2 | math_dapo |
MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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... | 1 | math_dapo |
MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 n... | 5 | math_dapo |
MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 $... | 8 | math_dapo |
MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 positi... | 107 | math_dapo |
MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 + ... | 24 | math_dapo |
MATH | {
"ground_truth": "3025",
"style": "rule-lighteval/MATH_v2"
} | Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
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 o... | 3025 | math_dapo |
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