data_source stringclasses 1 value | prompt listlengths 2 2 | ability stringclasses 1 value | reward_model dict | hint stringlengths 79 1.68k |
|---|---|---|---|---|
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of angle bisectors and the given trigonometric value. How might the angle bisector theorem and the Pythagorean theorem help you relate the sides of the triangle? What relationship can you infer from the given ratio $\frac{BD}{CD} = \frac{3}{2}$ and the right angle at $\angle BDC$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "113",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of a unit square and the formula for the area of a triangle. How might the positions of points $X$ and $Y$ on the perimeter affect the area of $\triangle AXY$? What relationship can you infer between the positions of $X$ and $Y$ and the expected value of the area? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of quadratic equations and their roots. How might the fact that the equations share a common root help you relate the coefficients $a, b,$ and $c$? What relationship can you infer from the common roots of the pairs of equations? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the pattern of card placement. How does the sequence of box numbers repeat, and what can you infer about the position of the last card based on the total number of cards and the repeating cycle? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_30k"
} | Think about the generating function approach. How might you use the properties of even and odd sums to simplify the expression? What relationship can you infer from the sum of the series involving powers of $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{5}$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of perfect squares. How might you express the given polynomial in a form that reveals its relationship to a perfect square? What happens if you try to complete the square or factor the polynomial? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationship between the arithmetic means of the sets \(\{a, b, c\}\) and \(\{a, b, c, d\}\). How can you express these means in terms of \(a, b, c,\) and \(d\)? What equation can you derive from the given condition? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the operations allowed. How do these operations affect the parity (odd/even) of the number of heads and tails? What relationship can you infer from the fact that the coins are in a circle? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "509",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of prime factorization in factorials. How might the prime factorization of $2018!$ relate to the powers of a prime $p$? What relationship can you infer from the condition that $2018!$ is divisible by $p^3$ but not by $p^4$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of a sphere tangent to a plane and the projection from a point. How might the given conic section $y = x^2$ relate to the position of the sphere and the point $P$? What relationship can you infer between the coordinates of $P$ and the radius of the sphere? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1007",
"style": "rule-lighteval/MATH_30k"
} | Think about the concept of graph connectivity. How might the number of edges in a complete graph relate to the number of travel companies? What is the minimum number of edges needed to ensure the graph remains connected after removing some edges? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the orthocenter and circumcenter in a triangle. How might the lengths of the sides of rectangle $HOMF$ relate to the triangle $\triangle ABC$? What relationship can you infer from the fact that $M$ is the midpoint of $BC$ and $F$ is the foot of the altitude from $A$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about Vieta's formulas. How might the sum and products of the roots relate to the coefficients of the polynomial? Also, consider the expression $(a-2)^3 + (b-2)^3 + (c-2)^3$. What identity or transformation could you use to simplify this expression in terms of $a$, $b$, and $c$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about the relative speeds and schedules of the buses. How might the frequency and speed of the buses affect Dirk's travel time? What relationship can you infer between the times the buses meet and Dirk's decision to switch buses? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationship between the area of a square and the area of a circle inscribed within it. How does the area change with each iteration of the process? What pattern can you identify in the areas of the black regions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_30k"
} | Think about the cost of each fruit. How might minimizing the cost of the more expensive fruits help you maximize the total number of fruits Danial could buy? What relationship can you infer from the total amount spent and the prices of the fruits? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the concept of expected value and how it applies to sequences. How might the condition of discarding rolls that result in consecutive terms summing to 5 affect the length of the sequence? What relationship can you infer between the probability of discarding a roll and the expected number of terms needed to include all four numbers? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "402",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the function $S(n)$. How might the relationship between $n$ and $S(n)$ help you narrow down the possible values of $n$? What can you infer about the factors of 2010 in this context? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the function $S(n)$ and how it changes when you add a number to $n$. What happens to the sum of the digits when you add a number that causes a carry-over in the digits? How might the value of $a$ influence the number of carry-overs? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about the Rational Root Theorem. How might the possible rational roots of the polynomial relate to the constant term and the leading coefficient? What can you infer about the factors of \(4k-8\) and how they might influence the integer roots? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "300",
"style": "rule-lighteval/MATH_30k"
} | Think about the distribution of candies as a function of the number of tasks solved. How might the distribution of candies relate to the number of students and the possible range of tasks solved? What is the maximum number of different performance levels there could be, and how does this affect the minimum number of candies needed? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of polynomial roots. How might the coefficients of the polynomials relate to the sum of their roots? What relationship can you infer from the given equations? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_30k"
} | Think about the generating function approach. How might you represent the sum of the series for each variable $a$, $b$, and $c$? What relationship can you infer from the condition that $a+b+c$ is even? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_30k"
} | Think about the possible outcomes of each move. What is the smallest number you can get from each type of move, and how does it relate to the initial numbers? What relationship can you infer from the operations performed on the numbers? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of prime divisors. How might the relationship between $n^2 + 3$ and $n^4 + 6$ help you identify the possible values of $n$? What can you infer about the prime divisors of these expressions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "8556",
"style": "rule-lighteval/MATH_30k"
} | Think about the expected value of the sum of a $k$-element subset of $n$ consecutive integers. How might the expected value of the die roll influence the overall expected value of the sum? What relationship can you infer between the expected value of the sum and the number of sides on the die? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_30k"
} | Think about the total number of cards each player needs to remove. How might the sequence of moves (1, 2, 3, 4 cards) influence the minimum number of cards required for the second player to complete both moves? What relationship can you infer from the sum of the cards removed by each player? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_30k"
} | Think about the total number of faces on all unit cubes. How might the expected number of painted faces change with each iteration of the process? What relationship can you infer between the number of iterations and the proportion of painted faces? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of tangents and radii in circles. How might the fact that $PQ = QR$ and the tangent at $P$ passes through $O$ help you relate the lengths and positions of these segments? What relationship can you infer from the tangency at point $A$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_30k"
} | Think about the definitions of arithmetic, geometric, and harmonic means. How might the conditions that these means are integers constrain the values of \(a\) and \(b\)? What relationship can you infer from the fact that the geometric mean is an integer? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_30k"
} | Think about the concept of winning and losing positions in combinatorial games. How might the number of stones taken in each turn influence the next player's options? What pattern can you observe in the sequence of moves that leads to a win for Dexter? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the incircle and the circumcircle. How might the given lengths and the fact that $DF = DA$ and $EF = EA$ help you relate the triangles $\triangle AED$ and $\triangle ABC$? What relationship can you infer from the circumradius of $\triangle AED$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of perfect squares. How might the expression $4^a + 2^b + 5$ relate to the form of a perfect square? What can you infer about the possible values of $a$ and $b$ by considering the parity and the range of the expression? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the structure of the set $M$. How does the pattern of numbers in $M$ relate to the sequence of all positive integers? What relationship can you infer between the sum of the elements and the number of terms, $2n$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of polynomials with integer coefficients. How might the number of distinct integer solutions to $P(x) = a$ vary as $a$ changes? What relationship can you infer between the degree of the polynomial and the number of distinct integer solutions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of exponents and how they can help you factor the expression. What relationship can you infer from the powers of 3 in the expression? How might you group the terms to reveal a common factor? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of modular arithmetic. How might the sum of the numbers from 1 to 2009 relate to the final numbers on the board when considered modulo 13? What relationship can you infer from the sum of the remaining numbers? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1099",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationship between the sum of the digits of a number and the number itself. How can you maximize the sum of the digits while minimizing the number? What happens to the ratio when the number is close to a power of 10? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of a regular graph and how the conditions given relate to the structure of the graph. How might the number of common acquaintances between pairs of people (whether they are acquainted or not) help you determine the possible values of \( n \)? What relationship can you infer from the given conditions about the degrees and the structure of the graph? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "40320",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of rooks and the structure of a chessboard. How does the color of the squares (black or white) influence the placement of the rooks? What relationship can you infer between the number of rows and columns and the number of rooks you need to place? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of polynomials and the Rational Root Theorem. How might the symmetry in the terms $(2 + x)^n$ and $(2 - x)^n$ influence the possible rational roots? What relationship can you infer from the equation when $x$ is a rational number? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about the symmetry of a regular 10-sided polygon. How might the properties of the diagonals and the center of the circle help you relate the positions of points $P$ and $Q$? What relationship can you infer from the intersection of the diagonals? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_30k"
} | Think about the Rational Root Theorem. How might it help you determine the possible rational roots of the polynomial? What conditions must \(m\) satisfy for all roots to be rational? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "270901",
"style": "rule-lighteval/MATH_30k"
} | Think about the concept of combinations with repetition. How might the total number of balls chosen from each box relate to the problem of distributing indistinguishable objects into distinct groups? What formula or method can you use to find the number of ways to distribute 900 balls into three groups where the order of selection does not matter? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of an equilateral triangle and the geometry of the circle. How might the symmetry of the equilateral triangle help you determine the direction and distance the boy swims? What relationship can you infer from the fact that $B$ is due west of $C$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_30k"
} | Think about the Power of a Point theorem. How might the lengths of the segments $AP$, $PB$, $CQ$, and $QD$ relate to the segments $PX$ and $PY$? What relationship can you infer from the given lengths and the theorem? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "4375",
"style": "rule-lighteval/MATH_30k"
} | Think about the identity for the difference of cubes, $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$. How might you use the given equations to find $x - y$ and $xy$? What relationship can you infer from the first equation to help you find $xy$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_30k"
} | Think about the geometric configuration that would allow each white point to have exactly two red points at a specific distance. How might the arrangement of these points (white and red) influence the minimum number of red points needed? What relationship can you infer from the given distance and the number of points? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the least common multiple (LCM) and greatest common divisor (GCD). How might the relationship between LCM and GCD of two numbers help you understand the equation? What special cases might you consider for \(m\) and \(n\) to explore the behavior of \(k\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of strongly connected graphs. How might the balance between in-degrees and out-degrees of vertices influence the connectivity of the graph? What relationship can you infer from the given condition about the difference between in-degrees and out-degrees? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the symmetry and structure of a regular tetrahedron. How might the division of edges into $n$ equal segments and the planes parallel to the faces affect the spatial arrangement of points? What relationship can you infer between the number of segments and the formation of a square-based rectangular parallelepiped? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2021",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of numbers that can be expressed as sums of powers. How might the constraints on $n$ and the number of terms (2021) influence the possible values of $m$? What relationship can you infer between the bases $m$ and $m+1$ when both are used to represent the same number $n$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationships between the sum of the integers and the sum of their squares. How might you use the given equations to express one variable in terms of the others? What can you infer about the possible values of $a$, $b$, and $c$ given their order and the equations? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationships between the equations. How might you use substitution or elimination to express one variable in terms of the others? What happens if you add or subtract the equations to simplify the system? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "132",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of prime factors. How might you test different values of \( n \) to see if \( n^2 - n + 11 \) has exactly four prime factors? What relationship can you infer from the form of the expression \( n^2 - n + 11 \)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1003",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of a regular polygon and the definition of a "good" diagonal. How might the parity of the number of sides in each segment created by a good diagonal influence the formation of isosceles triangles with two good sides? What relationship can you infer between the number of good diagonals and the number of such isosceles triangles? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_30k"
} | Think about the structure of the game and the sequence of events. How can you model the probability of a specific friend being called "falcon" twice before the game ends? What type of probability distribution might be useful here? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the divisors of $n$. How might the symmetry in the sums of rows and columns relate to the structure of the divisors? What special property must $n$ have for its divisors to fit these constraints? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationships between the fruits. How can you use the given exchange rates to find a relationship between pears and strawberries? What intermediate steps might you need to take to connect pears to strawberries through other fruits? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "80",
"style": "rule-lighteval/MATH_30k"
} | Think about the Pythagorean theorem. How might the lengths $AD$, $BD$, and $CD$ help you find the lengths of $AB$ and $AC$? What relationship can you infer from the extension of $AB$ to $E$ and the length $BE$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of prime numbers and perfect squares. How might the expression \(5^n - 1\) relate to these properties when divided by 3? What values of \(n\) could make the expression simpler to analyze? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of numbers that have exactly 4 positive divisors. How might the expression \(x^2 - x - 6\) factor, and what does this tell you about the possible values of \(x\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_30k"
} | Think about the structure of the grid and the placement of numbers. How might the requirement that numbers differing by 1 are adjacent influence the placement of numbers differing by 5000? What relationship can you infer between the positions of these numbers? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "152",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the incircle and the tangents drawn from a point to a circle. How might the fact that the incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same point $E$ influence the lengths of the segments $BD$ and $CD$? What relationship can you infer from the tangents drawn from $D$ to the incircles? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_30k"
} | Think about the condition given for a *good* arrangement. How might the relationship between the first and last numbers in any set of four consecutive numbers help you determine the values of the other numbers in the sequence? What can you infer about the relationship between the second and third numbers given that they are equal? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of odd natural numbers and their divisors. How might the number of divisors $d(n)$ relate to the largest divisor of $n$ that is different from $n$ itself? What special property does the number 9 have in this context? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about factoring the expression for \(A\). How might you rewrite \(A\) to reveal a pattern or a known form that could help you determine when it is a perfect square? What relationship can you infer from the structure of the polynomial? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of even and odd numbers. How might the sum of the integers (2023) influence the number of even and odd numbers in the set? What relationship can you infer between the number of even and odd numbers and the expression \(2d + 4m\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1987",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationship between the coefficients and the variables in the equation $ax + by + cz = 1$. How might the given values of $x$ and $z$ help you isolate $y$? What can you infer about the values of $a$, $b$, and $c$ given their constraints? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of quadratic polynomials. How might the given conditions $f(f(1)) = 0$ and $f(-f(-1)) = 0$ help you determine the roots of $f(x)$? What relationship can you infer between $f(1)$ and $-f(-1)$? Consider the form of a quadratic polynomial and how its roots and coefficients are related. |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_30k"
} | Think about the factorization of \(a^6 - 1\). How might the factors of \(a^6 - 1\) relate to the factors of \(a^3 - 1\) and \(a^2 - 1\)? What can you infer about the prime divisors of these expressions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationships between the given equations and the expression you need to find. How might you manipulate the equations to express $xy$ and $\frac{1}{xy}$ in terms of $x$ and $y$? What happens if you multiply the two given equations together? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "89",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of palindromic numbers. How might you use the structure of a palindrome to determine what $k$ needs to be when added to 25973? What is the smallest change you can make to 25973 to turn it into a palindrome? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the relationship between $a^m$ and $b^2$. What can you infer about the possible values of $m$ given that $a^m = b^2$ for all $a$ and $b$? How might the degrees of the polynomials $P(x)$ and $Q(x)$ relate to the degree of $R(x, y)$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of factorials and their relationship to trailing zeros. How might the number of trailing zeros in $n!(n+1)!(2n+1)!$ relate to the requirement that the last 30 digits of $n!(n+1)!(2n+1)! - 1$ are all 9? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_30k"
} | Think about the movement constraints of a rook. How can you break down the problem into smaller, manageable parts? What relationship can you infer between the number of moves and the positions the rook can return to? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of isosceles triangles and the sum of angles in a triangle. How might the given angles at point \( P \) help you find \( \angle BAP \)? What relationship can you infer from the angles around point \( P \)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "720",
"style": "rule-lighteval/MATH_30k"
} | Think about the geometry of the pentagon and the circles. How might the symmetry of the pentagon and the tangency of the circles relate to the path the sixth circle takes? What relationship can you infer from the angles in a regular pentagon and the rolling motion of the sixth circle? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of numbers divisible by both 37 and 173. What can you infer about the number's divisibility by their least common multiple? How might the thousands digit being 3 help you narrow down the possibilities for the hundreds digit? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3816",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the function $f(n)$. How might the condition $f(f(n)) = 3n$ help you understand the relationship between $f(n)$ and $n$? What can you infer about the growth rate of $f(n)$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of centroids and how they relate to the vertices of a triangle. How might the ratios given for the extensions of the sides of triangle $ABC$ affect the coordinates of the vertices of triangle $DEF$? What relationship can you infer between the centroids of triangles $ABC$ and $DEF$ based on these ratios? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_30k"
} | Think about the symmetry in the given equation and the condition \(x + y + z = 0\). How might you use the condition to simplify the expressions involving \(x, y,\) and \(z\)? What relationship can you infer from the given equation \(A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of prime numbers. How might the expressions \(4p^2 + 1\) and \(6p^2 + 1\) behave for different prime values of \(p\)? What relationship can you infer from the form of these expressions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_30k"
} | Think about the form of the equation $3 \times 2^x = n^2 - 1$. How can you rewrite $n^2 - 1$ to reveal a useful factorization? What properties of the factors might help you determine possible values of $x$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_30k"
} | Think about the symmetry in the equations. How might you use the symmetry to simplify the problem or find a relationship between \(x\), \(y\), and \(z\)? What happens if you add all three equations together? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of right angles and reflections. How might the reflection of point $C$ across line $BD$ help you understand the relationship between the angles at $A$ and $E$? What can you infer about the shape of the pentagon given the right angles at $A$ and $C$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1440",
"style": "rule-lighteval/MATH_30k"
} | Think about Vieta's formulas. How might the sum and product of the roots relate to the coefficients of the polynomial? What can you infer about the roots given that they are distinct negative integers? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "415",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of the semicircle inscribed in the triangle. How might the given ratio \(AD = 2DB\) help you relate the segments on \(AB\) and \(AC\)? What relationship can you infer between the radius of the semicircle and the segments \(AD\) and \(AE\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "936",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of a circle and the Pythagorean theorem. How might the radius of the circle and the segments $BD$ and $DC$ help you find the length of $AC$? What relationship can you infer from the right angle at $A$ and the circle centered at $A$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_30k"
} | Think about the divisibility rule for 11. How might the relationship between the digits of the number help you determine which digit $X$ makes the problem unsolvable? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_30k"
} | Think about the pattern of the number of haikus Ada writes each hour. How can you express this pattern mathematically? What relationship can you infer between the total number of haikus written and the hour number? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of equilateral triangles and how they affect the perimeter. How might the given angles and side lengths in the quadrilateral relate to the perimeter of the polygon \(APQRD\)? What relationship can you infer from the equation \(P(APQRD) = P(ABCD) + 32\)? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "6029",
"style": "rule-lighteval/MATH_30k"
} | Think about the geometric constraints of the problem. How does the diameter of the coin affect the regions on the checkerboard where the center of the coin can land without touching the lattice lines? What relationship can you infer from the dimensions of the checkerboard and the coin's diameter? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of quadratic functions and their tangents. How might the tangents to $G_2$ at points $A$ and $B$ relate to the quadratic function $G_1$? What can you infer about the slopes of these tangents and the coefficients of the quadratic functions? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of similar triangles and angle bisectors. How might the parallel lines and the angle bisector condition relate to the lengths of the segments in the triangle? What relationship can you infer from the given condition $A'C' + B'C' = AB$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_30k"
} | Think about the equation of a circle centered at the origin. How might the properties of prime numbers and the fact that $r$ is an odd integer help you narrow down the possible values of $r$? What relationship can you infer from the coordinates $(p^m, q^n)$ lying on the circle? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of numbers that end in zeroes. How might the sum of the digits of the nine-digit numbers relate to the divisibility rules for 10, 100, 1000, etc.? What relationship can you infer from the sum of the digits 1 through 9? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of inequalities and the relationships between the products and the individual numbers. How might the largest number in the set relate to the products of the other numbers? What happens if you consider the largest possible values for $a$, $b$, and $c$? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_30k"
} | Think about the system of linear equations. How might you use matrix operations or substitution to simplify and solve for \(x\)? What relationship can you infer from the coefficients of the variables in each equation? |
math_dapo_30k | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. 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\nRemember to put your answer on its own line after... | MATH | {
"ground_truth": "3025",
"style": "rule-lighteval/MATH_30k"
} | Think about the properties of perfect squares. How might the changes to the digits affect the new number's ability to be a perfect square? What relationship can you infer between the original and the new number? |
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