prompt stringlengths 41 2k | solution stringlengths 1 22 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | content listlengths 1 1 |
|---|---|---|---|---|---|---|---|
Let $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:
1. $x_1^2 + 5x_2^2 = 10$
2. $x_2y_1 - x_1y_2 = 5$
3. $x_1y_1 + 5x_2y_2 = \sqrt{105}$
Find the value of $y_1^2 + 5y_2^2$. | 23 | math_dapo | [
{
"content": "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\nLet $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:\n1. $x_1^2 + 5x_2^2 = 10$\n2. $x_2y_1 - x_1y_2 = 5$... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3eaf3bdc-3ea5-4df2-9630-2f7091931ed2"
} | [
{
"content": "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\nLet $x_1, x_2, y_1, y_2$ be real numbers satisfying the equations:\n1. $x_1^2 + 5x_2^2 = 10$\n2. $x_2y_1 - x_1y_2 = 5$... |
A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an equal number of adults and children will be seated in them. What is the least possible value of $N$? | 18 | math_dapo | [
{
"content": "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\nA single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "71e3b50a-ca98-4666-9ad7-93c4530d4924"
} | [
{
"content": "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\nA single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely filled, an... |
For any positive integer \(n\), let \[ f(n) = \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases} \] Find the value of \(\sum_{n = 1}^{1997}{f(n)}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 58 | math_dapo | [
{
"content": "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\nFor any positive integer \\(n\\), let \\[ f(n) = \\begin{cases} \\log_8{n}, & \\text{if }\\log_8{n}\\text{ is rational... | MATH | {
"ground_truth": "58",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f7f307fe-1705-4b18-8231-ce8b0b14cc50"
} | [
{
"content": "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\nFor any positive integer \\(n\\), let \\[ f(n) = \\begin{cases} \\log_8{n}, & \\text{if }\\log_8{n}\\text{ is rational... |
For $t = 1, 2, 3, 4$ , define $S_t = \sum\limits_{i = 1}^{350}a_i^t$ , where $a_i \in \{1,2,3,4\}$ . If $S_1 = 513$ and $S_4 = 4745$ , find the minimum possible value for $S_2$ . | 905 | math_dapo | [
{
"content": "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\nFor $t = 1, 2, 3, 4$ , define $S_t = \\sum\\limits_{i = 1}^{350}a_i^t$ , where $a_i \\in \\{1,2,3,4\\}$ . If $S_1 = 51... | MATH | {
"ground_truth": "905",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d82fc2b0-e6df-43e2-bc3d-5e3b6ae4008a"
} | [
{
"content": "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\nFor $t = 1, 2, 3, 4$ , define $S_t = \\sum\\limits_{i = 1}^{350}a_i^t$ , where $a_i \\in \\{1,2,3,4\\}$ . If $S_1 = 51... |
How many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? | 1 | math_dapo | [
{
"content": "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\nHow many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$?\n\nRemember to put your answer on... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d2dac85c-561b-4099-9fe0-626d2b783aa7"
} | [
{
"content": "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\nHow many positive real solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$?\n\nRemember to put your answer on... |
Suppose $f(x)$ is an invertible function, and suppose that $f(2)=f^{-1}(2)=4$.
What is the value of $f(f(2))$? | 2 | math_dapo | [
{
"content": "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\nSuppose $f(x)$ is an invertible function, and suppose that $f(2)=f^{-1}(2)=4$.\n\nWhat is the value of $f(f(2))$?\n\nR... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "352af3c4-ad7b-44b5-b11b-222f96f67334"
} | [
{
"content": "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\nSuppose $f(x)$ is an invertible function, and suppose that $f(2)=f^{-1}(2)=4$.\n\nWhat is the value of $f(f(2))$?\n\nR... |
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$? | 1 | math_dapo | [
{
"content": "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\nFor what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$?\n\nRemember to put your answer on its own line a... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "adeb69ee-6635-4e55-9b95-2dbabe1f6b28"
} | [
{
"content": "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\nFor what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$?\n\nRemember to put your answer on its own line a... |
A cube with side length $10$ is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three vertices adjacent to vertex $A$ are at heights $10, 11$, and $12$ above the plane. The distance from vertex $A$ to the plane can be expressed as $\frac{r-\sqrt{s}}{t}$ , where $r$ , $s$ , and $t$ are posi... | 330 | math_dapo | [
{
"content": "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\nA cube with side length $10$ is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three ve... | MATH | {
"ground_truth": "330",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3a6ed64f-7411-4313-9aba-6173ccddd6ee"
} | [
{
"content": "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\nA cube with side length $10$ is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three ve... |
For all integers $n \geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate
\[f(2) + f(3) + \cdots + f(100).\] | 124 | math_dapo | [
{
"content": "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\nFor all integers $n \\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\\sqrt[m]{n}$ is an intege... | MATH | {
"ground_truth": "124",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c19df7b7-3a57-447b-bf1c-9b7be81a0b9a"
} | [
{
"content": "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\nFor all integers $n \\geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\\sqrt[m]{n}$ is an intege... |
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
S(n)
= \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
+ 10 \left( n - 10^{\lfloor \log n \rfloor}
\cdot \... | 108 | math_dapo | [
{
"content": "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\nIf $x$ is a real number, let $\\lfloor x \\rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ i... | MATH | {
"ground_truth": "108",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "933c61d3-0386-4837-8d10-19da2e190f55"
} | [
{
"content": "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\nIf $x$ is a real number, let $\\lfloor x \\rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ i... |
Let $X$ be the sum of all divisors of the number $(3 \cdot 2009)^{((2 \cdot 2009)^{2009} - 1)}$. Find the last digit of $X$. | 0 | math_dapo | [
{
"content": "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\nLet $X$ be the sum of all divisors of the number $(3 \\cdot 2009)^{((2 \\cdot 2009)^{2009} - 1)}$. Find the last digit... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9261acb5-b0fa-45f3-8fd1-60ece6c309cb"
} | [
{
"content": "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\nLet $X$ be the sum of all divisors of the number $(3 \\cdot 2009)^{((2 \\cdot 2009)^{2009} - 1)}$. Find the last digit... |
Find the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$. | 98 | math_dapo | [
{
"content": "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\nFind the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of... | MATH | {
"ground_truth": "98",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e877fd8b-7b52-430f-bcb7-d6fb497185ad"
} | [
{
"content": "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\nFind the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of... |
A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\sqrt{m}$, please give the value of $k + m$. | 3 | math_dapo | [
{
"content": "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\nA circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\\sq... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "46a72764-62f4-4d5e-95eb-2e2b9b98f1eb"
} | [
{
"content": "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\nA circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\\sq... |
Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral?
\begin{asy}
size(4cm);
fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3));
draw((0,0)--(0,12)--(-5,0)--cycle);
draw((0,0)--(8,0)--(0,6));
label("5", (-2.5,0), S);
label("13", (-2.5,... | 42 | math_dapo | [
{
"content": "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\nTwo right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadr... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9f3d6f12-2f97-4cf0-87f5-46d8e408a73a"
} | [
{
"content": "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\nTwo right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadr... |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | 15 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA list of five positive integers has all of the following properties:\n\n$\\bullet$ The only integer in the list that... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e1579fbc-c150-43ad-97ca-0b61b6e3d619"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA list of five positive integers has all of the following properties:\n\n$\\bullet$ The only integer in the list that... |
The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? | 36 | math_dapo | [
{
"content": "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\nThe product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of t... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b0e13737-4704-4a06-a473-5c4b0ddbaa37"
} | [
{
"content": "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\nThe product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of t... |
Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, whe... | 87 | math_dapo | [
{
"content": "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\nLet $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisect... | MATH | {
"ground_truth": "87",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2f6f7fc5-c3fe-4cb3-9a7c-c4b7c6017c42"
} | [
{
"content": "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\nLet $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisect... |
Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices? | 808500 | math_dapo | [
{
"content": "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\nCaltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be fo... | MATH | {
"ground_truth": "808500",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "758fba91-cb06-4570-964b-3806ed16ed9d"
} | [
{
"content": "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\nCaltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be fo... |
A cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$. | 13 | math_dapo | [
{
"content": "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\nA cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle c... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "31f55c2c-9c49-435d-a64c-82ac6d4dcb0a"
} | [
{
"content": "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\nA cyclic quadrilateral $ABCD$ has side lengths $AB = 3$, $BC = AD = 5$, and $CD = 8$. The radius of its circumcircle c... |
Let $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\] | 6 | math_dapo | [
{
"content": "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\nLet $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of\n\\[6a^3 + 9b^3 + 32c^3 + \\frac{1}{4... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6da953ee-fc85-4f96-8a52-04bb33f00d96"
} | [
{
"content": "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\nLet $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of\n\\[6a^3 + 9b^3 + 32c^3 + \\frac{1}{4... |
A trifecta is an ordered triple of positive integers \((a, b, c)\) with \(a < b < c\) such that \(a\) divides \(b\), \(b\) divides \(c\), and \(c\) divides \(ab\). What is the largest possible sum \(a + b + c\) over all trifectas of three-digit integers? | 1736 | math_dapo | [
{
"content": "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\nA trifecta is an ordered triple of positive integers \\((a, b, c)\\) with \\(a < b < c\\) such that \\(a\\) divides \\... | MATH | {
"ground_truth": "1736",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f92e748b-1c67-42b0-a658-b6565e366583"
} | [
{
"content": "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\nA trifecta is an ordered triple of positive integers \\((a, b, c)\\) with \\(a < b < c\\) such that \\(a\\) divides \\... |
The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. | 9 | math_dapo | [
{
"content": "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\nThe decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digit... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "531cd0c7-8664-48d6-b0f4-71d5dd2e5a99"
} | [
{
"content": "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\nThe decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digit... |
In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player h... | 4 | math_dapo | [
{
"content": "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\nIn a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win fo... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4c3a44db-da3d-4e20-a76d-9d0b0568982c"
} | [
{
"content": "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\nIn a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win fo... |
Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$. There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$. Compute the distance between the centers of these two circles.The answer is in the form k\sqrt{m}+n,. Please p... | 18 | math_dapo | [
{
"content": "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\nLet $\\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$. There are two circles located inside $\\angle B... | MATH | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f454b064-64ce-4dd9-b7eb-c964fbcf2204"
} | [
{
"content": "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\nLet $\\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$. There are two circles located inside $\\angle B... |
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:
\[
1 < a < b+2 < 10.
\] | 28 | math_dapo | [
{
"content": "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\nFind the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:\n\n\\[\n1 < a < b+2 < 10.\n\\]\n\nR... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d82f3f59-03c4-4a43-be02-953578622ec4"
} | [
{
"content": "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\nFind the number of ordered pairs of integers $(a, b)$ that satisfy the inequality:\n\n\\[\n1 < a < b+2 < 10.\n\\]\n\nR... |
Determine the largest prime which divides both $2^{24} - 1$ and $2^{16} - 1$. | 17 | math_dapo | [
{
"content": "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\nDetermine the largest prime which divides both $2^{24} - 1$ and $2^{16} - 1$.\n\nRemember to put your answer on its ow... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "887adca6-5d52-46a8-a939-b5c2da641892"
} | [
{
"content": "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\nDetermine the largest prime which divides both $2^{24} - 1$ and $2^{16} - 1$.\n\nRemember to put your answer on its ow... |
Points $A = (3,9), B = (1,1), C = (5,3),$ and $D = (a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA}$ is a square. Find the sum of the coordinates of point $D$. | 10 | math_dapo | [
{
"content": "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\nPoints $A = (3,9), B = (1,1), C = (5,3),$ and $D = (a,b)$ lie in the first quadrant and are the vertices of quadrilate... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8d2432ee-3fa2-4599-91ea-0b7f844de6aa"
} | [
{
"content": "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\nPoints $A = (3,9), B = (1,1), C = (5,3),$ and $D = (a,b)$ lie in the first quadrant and are the vertices of quadrilate... |
S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$. | 24 | math_dapo | [
{
"content": "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\nS is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + ... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "134e2cee-3209-4fac-bd6f-f1981ee56708"
} | [
{
"content": "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\nS is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + ... |
Find the value of $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$. The original answer is in the form $-x*2^{-(y*z+k)}$, where x, y, and z are integers. Please provide the sum of x, y, and z. | 4 | math_dapo | [
{
"content": "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\nFind the value of $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$. The original answer is in the form $-x*2^{-(y*z+k)}$, where x... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6e9b5edc-63d4-4557-ba5b-e29d368b69ea"
} | [
{
"content": "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\nFind the value of $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$. The original answer is in the form $-x*2^{-(y*z+k)}$, where x... |
The remainder of $1991^{2000}$ modulo $10^6$ is \underline{\hspace{2cm}}. | 1 | math_dapo | [
{
"content": "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\nThe remainder of $1991^{2000}$ modulo $10^6$ is \\underline{\\hspace{2cm}}.\n\nRemember to put your answer on its own ... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c6c00e82-0255-4944-a1a2-1524249d2abd"
} | [
{
"content": "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\nThe remainder of $1991^{2000}$ modulo $10^6$ is \\underline{\\hspace{2cm}}.\n\nRemember to put your answer on its own ... |
Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\left(\sqrt{... | 803 | math_dapo | [
{
"content": "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\nPatio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with... | MATH | {
"ground_truth": "803",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fd9e70b0-8df3-4681-a76b-61e958660c16"
} | [
{
"content": "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\nPatio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with... |
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$. | 17 | math_dapo | [
{
"content": "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\nLet $(a_n)_{n \\geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a1b445e5-08a4-4043-8177-7fdfea324a8e"
} | [
{
"content": "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\nLet $(a_n)_{n \\geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+... |
On Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types: some always lie, some always tell the truth, and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statem... | 7 | math_dapo | [
{
"content": "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\nOn Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "95602b93-13f8-4813-855e-7b1dd5e3174d"
} | [
{
"content": "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\nOn Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types... |
Let $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\{ 1, 2, 3, \ldots , n\}$ can be partitioned into $3$ subsets with equal sums. Find the number of elements of $E$. | 64 | math_dapo | [
{
"content": "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\nLet $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\\{ 1, 2, 3, \\ldots , n\\}$ ca... | MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "70275159-52dc-4d3c-8e81-d7a7dae3bf96"
} | [
{
"content": "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\nLet $E$ denote the set of all natural numbers $n$ such that $3 < n < 100$ and the set $\\{ 1, 2, 3, \\ldots , n\\}$ ca... |
Triangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n... | 3308 | math_dapo | [
{
"content": "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\nTriangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it ... | MATH | {
"ground_truth": "3308",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8e322f37-708b-4050-86a6-4b1569660dfc"
} | [
{
"content": "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\nTriangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it ... |
Find $n$ such that $20^{2009}=10^{2000}\cdot 40^9\cdot 2^n$. | 1991 | math_dapo | [
{
"content": "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\nFind $n$ such that $20^{2009}=10^{2000}\\cdot 40^9\\cdot 2^n$.\n\nRemember to put your answer on its own line after \"... | MATH | {
"ground_truth": "1991",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b5e85ba7-4292-4bbb-a299-89ac40ac4924"
} | [
{
"content": "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\nFind $n$ such that $20^{2009}=10^{2000}\\cdot 40^9\\cdot 2^n$.\n\nRemember to put your answer on its own line after \"... |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y \geq A$ where $0 \leq x \leq 6$ and $0 \leq y \leq 7$. | 10 | math_dapo | [
{
"content": "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\nFind the natural number $A$ such that there are $A$ integer solutions to $x+y \\geq A$ where $0 \\leq x \\leq 6$ and $... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "77dd3414-9327-42a4-93ec-1debf496d65e"
} | [
{
"content": "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\nFind the natural number $A$ such that there are $A$ integer solutions to $x+y \\geq A$ where $0 \\leq x \\leq 6$ and $... |
We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 bl... | 859 | math_dapo | [
{
"content": "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\nWe colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that ... | MATH | {
"ground_truth": "859",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "068bb15c-4bea-4ad3-917f-9dd188230f74"
} | [
{
"content": "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\nWe colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that ... |
What is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 8 | math_dapo | [
{
"content": "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\nWhat is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc412e12-5e3e-4752-9ce0-4cb991c48adf"
} | [
{
"content": "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\nWhat is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express... |
The four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it vanishes, and a number of copies equal to the number on the face it landed on appear. For example:
- If it lands on face $0,$ it disappears.
- If it lands on face $1,$ nothing happens.
- If it lands on f... | 24 | math_dapo | [
{
"content": "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\nThe four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it v... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8684569d-ee95-4f32-bb8b-97cb350e65c7"
} | [
{
"content": "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\nThe four faces of a tetrahedral die are labeled $0, 1, 2,$ and $3.$ The die has a peculiar property: when rolled, it v... |
Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$. | 7 | math_dapo | [
{
"content": "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\nFind the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.\n\nRemember to put your... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "93f6fede-105d-4e77-ab07-58624416a412"
} | [
{
"content": "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\nFind the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.\n\nRemember to put your... |
The weight of \( \frac{1}{3} \) of a large pizza together with \( 3 \frac{1}{2} \) cups of orange slices is the same as the weight of \( \frac{3}{4} \) of a large pizza together with \( \frac{1}{2} \) cup of orange slices. A cup of orange slices weighs \( \frac{1}{4} \) of a pound. What is the weight, in pounds, of a l... | 14 | math_dapo | [
{
"content": "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\nThe weight of \\( \\frac{1}{3} \\) of a large pizza together with \\( 3 \\frac{1}{2} \\) cups of orange slices is the ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ea32508b-2905-4e5d-acde-6d26978e989c"
} | [
{
"content": "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\nThe weight of \\( \\frac{1}{3} \\) of a large pizza together with \\( 3 \\frac{1}{2} \\) cups of orange slices is the ... |
Let $2^{1110} \equiv n \pmod{1111}$ with $0 \leq n < 1111$. Compute $n$. | 1024 | math_dapo | [
{
"content": "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\nLet $2^{1110} \\equiv n \\pmod{1111}$ with $0 \\leq n < 1111$. Compute $n$.\n\nRemember to put your answer on its own ... | MATH | {
"ground_truth": "1024",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "65809813-1d2a-407c-9cba-b10c1419184a"
} | [
{
"content": "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\nLet $2^{1110} \\equiv n \\pmod{1111}$ with $0 \\leq n < 1111$. Compute $n$.\n\nRemember to put your answer on its own ... |
The value of the expression $\frac{(304)^5}{(29.7)(399)^4}$ is closest to what integer? | 3 | math_dapo | [
{
"content": "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\nThe value of the expression $\\frac{(304)^5}{(29.7)(399)^4}$ is closest to what integer?\n\nRemember to put your answe... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "13417584-f9c3-49b2-ad9c-ff38036bf34d"
} | [
{
"content": "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\nThe value of the expression $\\frac{(304)^5}{(29.7)(399)^4}$ is closest to what integer?\n\nRemember to put your answe... |
Find the number of solutions to
\[\frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta} - \frac{1}{\tan^2 \theta} - \frac{1}{\cot^2 \theta} - \frac{1}{\sec^2 \theta} - \frac{1}{\csc^2 \theta} = -3\]in the interval $0 \le \theta \le 2 \pi.$ | 4 | math_dapo | [
{
"content": "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\nFind the number of solutions to\n\\[\\frac{1}{\\sin^2 \\theta} - \\frac{1}{\\cos^2 \\theta} - \\frac{1}{\\tan^2 \\thet... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "aeab3d37-88d4-49d7-a6f1-0817538a814e"
} | [
{
"content": "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\nFind the number of solutions to\n\\[\\frac{1}{\\sin^2 \\theta} - \\frac{1}{\\cos^2 \\theta} - \\frac{1}{\\tan^2 \\thet... |
In the sequence $2001$, $2002$, $2003$, $\cdots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the $2004^{th}$ term in this sequence? | 0 | math_dapo | [
{
"content": "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\nIn the sequence $2001$, $2002$, $2003$, $\\cdots$, each term after the third is found by subtracting the previous term... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d6a10521-1464-4f52-9eec-370752c04421"
} | [
{
"content": "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\nIn the sequence $2001$, $2002$, $2003$, $\\cdots$, each term after the third is found by subtracting the previous term... |
Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$. | 335 | math_dapo | [
{
"content": "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\nCompute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.\n\nRemember to put your answer o... | MATH | {
"ground_truth": "335",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "88dfaf3f-23fe-4a3c-9e1b-0fbc710178a0"
} | [
{
"content": "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\nCompute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.\n\nRemember to put your answer o... |
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$. | 33 | math_dapo | [
{
"content": "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\nA number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fd787c35-09d5-4ab2-b82e-561594e435b4"
} | [
{
"content": "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\nA number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine... |
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S \cup \{15\}$ and $S \cup \{15, 1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have? | 4 | math_dapo | [
{
"content": "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\nLet $S$ be a set of real numbers with mean $M$. If the means of the sets $S \\cup \\{15\\}$ and $S \\cup \\{15, 1\\}$ ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3c40c454-5aa5-48b0-8fc8-7002c9a38c4c"
} | [
{
"content": "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\nLet $S$ be a set of real numbers with mean $M$. If the means of the sets $S \\cup \\{15\\}$ and $S \\cup \\{15, 1\\}$ ... |
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$. | 11 | math_dapo | [
{
"content": "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\nFind the least positive integer that cannot be represented as $\\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bce80766-034b-4206-9666-dc161818d0cf"
} | [
{
"content": "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\nFind the least positive integer that cannot be represented as $\\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a... |
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please giv... | 1171 | math_dapo | [
{
"content": "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\nTwo different cubes of the same size are to be painted, with the color of each face being chosen independently and at ... | MATH | {
"ground_truth": "1171",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f92eb769-ef01-4376-a8b6-2e00a8a6c76a"
} | [
{
"content": "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\nTwo different cubes of the same size are to be painted, with the color of each face being chosen independently and at ... |
The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;
for(int i = -4; i <= 5; ++i) {
draw((i,-1)--(i,6), dashed+med... | 2 | math_dapo | [
{
"content": "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\nThe graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]\nimport graph; si... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "52fa8420-e814-4bdf-b633-29c72529a5ae"
} | [
{
"content": "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\nThe graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]\nimport graph; si... |
In triangle $ABC$, $\angle ABC = 90^\circ$ and $AD$ is an angle bisector. If $AB = 90,$ $BC = x$, and $AC = 2x - 6,$ then find the area of $\triangle ADC$. Round your answer to the nearest integer. | 1363 | math_dapo | [
{
"content": "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\nIn triangle $ABC$, $\\angle ABC = 90^\\circ$ and $AD$ is an angle bisector. If $AB = 90,$ $BC = x$, and $AC = 2x - 6,... | MATH | {
"ground_truth": "1363",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e4015ea5-2d3f-4cbe-8b39-5c6c991a70d6"
} | [
{
"content": "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\nIn triangle $ABC$, $\\angle ABC = 90^\\circ$ and $AD$ is an angle bisector. If $AB = 90,$ $BC = x$, and $AC = 2x - 6,... |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$? | 3 | math_dapo | [
{
"content": "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\nLet $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "70efc135-5c5e-41d8-b1ee-f16a09ed63f2"
} | [
{
"content": "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\nLet $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<... |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? Provide your answer as an integer. | 8 | math_dapo | [
{
"content": "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\nWhat is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $1... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fd2f1661-d32d-472e-a031-d96b9b606cfa"
} | [
{
"content": "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\nWhat is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $1... |
In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees. | 30 | math_dapo | [
{
"content": "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\nIn the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polyg... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28969820-55ae-4fb3-9647-ebdba5b772fa"
} | [
{
"content": "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\nIn the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polyg... |
Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three feet of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b... | 600 | math_dapo | [
{
"content": "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\nPoint $P$ is in the interior of $\\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The thre... | MATH | {
"ground_truth": "600",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bbe33e70-b80c-4a9f-9cd6-3e303e0a265d"
} | [
{
"content": "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\nPoint $P$ is in the interior of $\\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The thre... |
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, an... | 280 | math_dapo | [
{
"content": "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\nLet $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\\... | MATH | {
"ground_truth": "280",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "82187f1e-f4e9-4a78-9229-5fd7f012b56e"
} | [
{
"content": "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\nLet $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\\... |
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizont... | 60 | math_dapo | [
{
"content": "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\nA $ 3\\times3\\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of ... | MATH | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e04da163-aab6-4bb5-9af2-6f2450bebe9a"
} | [
{
"content": "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\nA $ 3\\times3\\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of ... |
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. [asy]
size(250);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
dr... | 6 | math_dapo | [
{
"content": "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\nIn the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sid... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f66c778-bf05-40e0-923c-a19c1e5e443c"
} | [
{
"content": "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\nIn the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sid... |
The function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all positive integers $n.$ Find $f(1000).$ | 1016 | math_dapo | [
{
"content": "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\nThe function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all posi... | MATH | {
"ground_truth": "1016",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8776775d-f917-4ab8-83ae-6e4ed722e0e2"
} | [
{
"content": "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\nThe function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all posi... |
Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass. | 4 | math_dapo | [
{
"content": "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\nSix lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.\n\nR... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "39ce29ee-85e0-4c42-b6e8-bc1e8b2b9b1f"
} | [
{
"content": "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\nSix lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.\n\nR... |
A triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle? | 76 | math_dapo | [
{
"content": "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\nA triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle?\n\nReme... | MATH | {
"ground_truth": "76",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a7ad5369-5175-43e8-8de3-5da8b4fce355"
} | [
{
"content": "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\nA triangle has an area of 114 square units and sides of integer length. What is the perimeter of the triangle?\n\nReme... |
If $\frac{60}{2^3\cdot5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point? | 3 | math_dapo | [
{
"content": "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\nIf $\\frac{60}{2^3\\cdot5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal poin... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28ed7abf-f79b-4171-8386-3d1bd28316c3"
} | [
{
"content": "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\nIf $\\frac{60}{2^3\\cdot5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal poin... |
Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$. | 99 | math_dapo | [
{
"content": "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\nFind the number of integers $n$ such that $1 \\le n \\le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.\n\n... | MATH | {
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "48446bbe-a5c8-4002-a9b1-b4128978c08d"
} | [
{
"content": "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\nFind the number of integers $n$ such that $1 \\le n \\le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.\n\n... |
Positive integers $a$, $b$, and $c$ satisfy the equation:
\[
\frac{ab}{a - b} = c
\]
What is the largest possible value of $a + b + c$ that does not exceed 99? | 99 | math_dapo | [
{
"content": "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\nPositive integers $a$, $b$, and $c$ satisfy the equation:\n\\[\n\\frac{ab}{a - b} = c\n\\]\nWhat is the largest possib... | MATH | {
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c64e605e-ebde-4ff8-a8bf-25af62abe96a"
} | [
{
"content": "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\nPositive integers $a$, $b$, and $c$ satisfy the equation:\n\\[\n\\frac{ab}{a - b} = c\n\\]\nWhat is the largest possib... |
Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\). | 427 | math_dapo | [
{
"content": "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\nLet \\(ABCD\\) be a cyclic quadrilateral. The side lengths of \\(ABCD\\) are distinct integers less than \\(15\\) such... | MATH | {
"ground_truth": "427",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8fb5107d-11e7-4f7d-a938-0335be0a710e"
} | [
{
"content": "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\nLet \\(ABCD\\) be a cyclic quadrilateral. The side lengths of \\(ABCD\\) are distinct integers less than \\(15\\) such... |
In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-s... | 8 | math_dapo | [
{
"content": "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\nIn regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\\overline{BC}$ and $\\overline{FG}$ respectively. Compu... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7a0e53e0-c246-44e4-a51e-d4cca3888233"
} | [
{
"content": "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\nIn regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\\overline{BC}$ and $\\overline{FG}$ respectively. Compu... |
Let $S$ be a set with six elements. Let $\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $\mathcal{P}$. The probability that $B$ is contained in one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive... | 710 | math_dapo | [
{
"content": "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\nLet $S$ be a set with six elements. Let $\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, n... | MATH | {
"ground_truth": "710",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bc56133a-987e-43ef-86e6-06e0c0944836"
} | [
{
"content": "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\nLet $S$ be a set with six elements. Let $\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, n... |
A number is called *purple* if it can be expressed in the form $\frac{1}{2^a 5^b}$ for positive integers $a > b$. The sum of all purple numbers can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. | 109 | math_dapo | [
{
"content": "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\nA number is called *purple* if it can be expressed in the form $\\frac{1}{2^a 5^b}$ for positive integers $a > b$. The... | MATH | {
"ground_truth": "109",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d6b38688-a76d-4642-8168-38c7b788bab9"
} | [
{
"content": "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\nA number is called *purple* if it can be expressed in the form $\\frac{1}{2^a 5^b}$ for positive integers $a > b$. The... |
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
[asy] import three; import solids; size(5cm); currentprojection=o... | 384 | math_dapo | [
{
"content": "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\nA cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ fee... | MATH | {
"ground_truth": "384",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3504f4cc-6838-427e-b2e7-7952663f98dc"
} | [
{
"content": "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\nA cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ fee... |
On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is $1088$ feet per second and one mile is $5280$ feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. If the answer is $x$ miles, what is the valu... | 2 | math_dapo | [
{
"content": "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\nOn a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. ... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2ea2ffd6-23f2-40c2-be2c-d3883337ebec"
} | [
{
"content": "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\nOn a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. ... |
There are exactly two four-digit numbers that are multiples of three where their first digit is double their second digit, their third digit is three more than their fourth digit, and their second digit is $2$ less than their fourth digit. Find the difference of these two numbers. | 6333 | math_dapo | [
{
"content": "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\nThere are exactly two four-digit numbers that are multiples of three where their first digit is double their second di... | MATH | {
"ground_truth": "6333",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a7a1e20b-8b7b-4bfd-bf23-72d41306ac8e"
} | [
{
"content": "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\nThere are exactly two four-digit numbers that are multiples of three where their first digit is double their second di... |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the five of them to sit in a row of $5$ chairs under these conditions. | 28 | math_dapo | [
{
"content": "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\nAlice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the f... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2533c8a2-2ad4-4f24-9a1f-3110925c3b2e"
} | [
{
"content": "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\nAlice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the f... |
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(... | 288 | math_dapo | [
{
"content": "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\nThe solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All ... | MATH | {
"ground_truth": "288",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "652109fe-7626-4d05-8a47-2c7c6caad3c3"
} | [
{
"content": "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\nThe solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All ... |
Values for $A,B,C,$ and $D$ are to be selected from $\{1, 2, 3, 4, 5, 6\}$ without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves $y=Ax^2+B$ and $y=Cx^2+D$ intersect? (The order in which the curves are listed does not matter; for example, the c... | 90 | math_dapo | [
{
"content": "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\nValues for $A,B,C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters ... | MATH | {
"ground_truth": "90",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f0d0f112-9e57-4fcf-8222-294ce7b221ee"
} | [
{
"content": "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\nValues for $A,B,C,$ and $D$ are to be selected from $\\{1, 2, 3, 4, 5, 6\\}$ without replacement (i.e. no two letters ... |
One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). If the postage for a letter weighing $4.5$ ounces is x cents, what is the value of x? | 118 | math_dapo | [
{
"content": "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\nOne proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ... | MATH | {
"ground_truth": "118",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b2fa9b0f-c97c-4d5c-a5ac-c1a6e259ab00"
} | [
{
"content": "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\nOne proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ... |
If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, find the value of $a+b$. | -2 | math_dapo | [
{
"content": "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\nIf $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, find the value of $a+b$.\n\nRemember to put your answer... | MATH | {
"ground_truth": "-2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "46e78d90-e93f-4e61-ab93-2d32dada10be"
} | [
{
"content": "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\nIf $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, find the value of $a+b$.\n\nRemember to put your answer... |
How many interior diagonals does an icosahedron have? (An $\emph{icosahedron}$ is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex. An $\emph{interior}$ diagonal is a segment connecting two vertices which do not lie on a common face.) | 36 | math_dapo | [
{
"content": "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\nHow many interior diagonals does an icosahedron have? (An $\\emph{icosahedron}$ is a 3-dimensional figure with 20 tri... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c573ab66-0d0b-42fa-a41e-774c1cc4103f"
} | [
{
"content": "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\nHow many interior diagonals does an icosahedron have? (An $\\emph{icosahedron}$ is a 3-dimensional figure with 20 tri... |
Steve guesses randomly on a 15-question multiple choice test in which each problem has two choices, a correct one and an incorrect one. What is the probability that he gets at least half of the questions correct? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please ... | 3 | math_dapo | [
{
"content": "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\nSteve guesses randomly on a 15-question multiple choice test in which each problem has two choices, a correct one and ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de3ce0c3-fccc-4436-b607-4a54f5eec99d"
} | [
{
"content": "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\nSteve guesses randomly on a 15-question multiple choice test in which each problem has two choices, a correct one and ... |
Let $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC$. The expression $\frac{[ABH']}{[ACH']}$ can be written as $\frac{p}{q}$. Find $p+q$. | 251 | math_dapo | [
{
"content": "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\nLet $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\\triangle ABC$ a... | MATH | {
"ground_truth": "251",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ef960371-7ff6-43e4-aa25-5af7aa51a1c5"
} | [
{
"content": "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\nLet $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\\triangle ABC$ a... |
Suppose that $g(x)=f^{-1}(x)$. If $g(-15)=0$, $g(0)=3$, $g(3)=9$ and $g(9)=20$, what is $f(f(9))$? | 0 | math_dapo | [
{
"content": "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\nSuppose that $g(x)=f^{-1}(x)$. If $g(-15)=0$, $g(0)=3$, $g(3)=9$ and $g(9)=20$, what is $f(f(9))$?\n\nRemember to put ... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7e6be3c6-d12a-4994-b257-f9d1bd56a8fe"
} | [
{
"content": "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\nSuppose that $g(x)=f^{-1}(x)$. If $g(-15)=0$, $g(0)=3$, $g(3)=9$ and $g(9)=20$, what is $f(f(9))$?\n\nRemember to put ... |
How many integers between 100 and 500 have at least two 3s as digits? | 22 | math_dapo | [
{
"content": "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\nHow many integers between 100 and 500 have at least two 3s as digits?\n\nRemember to put your answer on its own line a... | MATH | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e7e49d1c-c6e7-4982-bfb5-ee79ad489665"
} | [
{
"content": "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\nHow many integers between 100 and 500 have at least two 3s as digits?\n\nRemember to put your answer on its own line a... |
A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal ... | 10 | math_dapo | [
{
"content": "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\nA girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will ins... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "61274a14-c2a2-4843-94bd-eced174f3d9f"
} | [
{
"content": "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\nA girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will ins... |
A polynomial $p$ can be written as
\begin{align*}
p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.
\end{align*}
Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$. | 64 | math_dapo | [
{
"content": "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\nA polynomial $p$ can be written as \n\\begin{align*}\np(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.\n\\end{align*}\nGiven that ... | MATH | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d669e20f-ff1a-4fd2-a92b-dd64ad4feb0e"
} | [
{
"content": "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\nA polynomial $p$ can be written as \n\\begin{align*}\np(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.\n\\end{align*}\nGiven that ... |
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expresse... | 31 | math_dapo | [
{
"content": "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\nLet triangle $ABC$ with incenter $I$ and circumcircle $\\Gamma$ satisfy $AB = 6\\sqrt{3}$, $BC = 14$, and $CA = 22$. C... | MATH | {
"ground_truth": "31",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "55935d4a-741c-491c-ab4e-c6878f8ab41d"
} | [
{
"content": "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\nLet triangle $ABC$ with incenter $I$ and circumcircle $\\Gamma$ satisfy $AB = 6\\sqrt{3}$, $BC = 14$, and $CA = 22$. C... |
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex.... | 2040200 | math_dapo | [
{
"content": "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\nTessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real num... | MATH | {
"ground_truth": "2040200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "400f43b0-879f-4158-a88e-b8593004224d"
} | [
{
"content": "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\nTessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real num... |
Simplify
\[\frac{1}{\log_{15} 2 + 1} + \frac{1}{\log_{10} 3 + 1} + \frac{1}{\log_6 5 + 1}.\] | 2 | math_dapo | [
{
"content": "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\nSimplify\n\\[\\frac{1}{\\log_{15} 2 + 1} + \\frac{1}{\\log_{10} 3 + 1} + \\frac{1}{\\log_6 5 + 1}.\\]\n\nRemember to p... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4b180bf8-8d1d-48aa-bc32-fe5e2f5c847f"
} | [
{
"content": "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\nSimplify\n\\[\\frac{1}{\\log_{15} 2 + 1} + \\frac{1}{\\log_{10} 3 + 1} + \\frac{1}{\\log_6 5 + 1}.\\]\n\nRemember to p... |
Ten people are sitting around a round table. Three of them are chosen at random to give a presentation. What is the probability that the three chosen people were sitting in consecutive seats?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 13 | math_dapo | [
{
"content": "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\nTen people are sitting around a round table. Three of them are chosen at random to give a presentation. What is the ... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0d2c211d-9244-435f-b3d7-ec8c0d30aa9a"
} | [
{
"content": "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\nTen people are sitting around a round table. Three of them are chosen at random to give a presentation. What is the ... |
Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value o... | 109 | math_dapo | [
{
"content": "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\nRyan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turn... | MATH | {
"ground_truth": "109",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5416dce9-acd6-439f-a55e-f33ccc375828"
} | [
{
"content": "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\nRyan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turn... |
For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day $0$. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree. For each sapling, she succeeds in turning it into a ... | 789953 | math_dapo | [
{
"content": "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\nFor her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six t... | MATH | {
"ground_truth": "789953",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c313ae87-7556-479a-9fec-889db4f8828c"
} | [
{
"content": "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\nFor her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six t... |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | 2 | math_dapo | [
{
"content": "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\nFor $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29823a97-d67b-4539-a573-0bcb54de8894"
} | [
{
"content": "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\nFor $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7... |
Find all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square. | 3 | math_dapo | [
{
"content": "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\nFind all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square.\n\nRemember to put your answer on its own li... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1034e651-8205-4668-bafe-072c722b1644"
} | [
{
"content": "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\nFind all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square.\n\nRemember to put your answer on its own li... |
Given a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible ways to dissect this polygon into 12 equilateral triangles and 6 squares. Let $T_1$ denote the union of all triangles in the first dissection, and $S_1$ the union of all squares. Define $T_2$ and $S_2$ si... | 40003 | math_dapo | [
{
"content": "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\nGiven a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible wa... | MATH | {
"ground_truth": "40003",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "202b27f8-5cbe-43b2-9502-dd01826b5279"
} | [
{
"content": "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\nGiven a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible wa... |
Triangle $\triangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $BC$, and suppose $AD = 12$. If $BD = \frac{1}{4} BC$ and $OH \parallel BC$, compute $AB^2$. | 160 | math_dapo | [
{
"content": "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\nTriangle $\\triangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $B... | MATH | {
"ground_truth": "160",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9cf161af-5c57-4f81-8133-6ec0a98f25d1"
} | [
{
"content": "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\nTriangle $\\triangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $B... |
Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$. | 6 | math_dapo | [
{
"content": "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\nEvery cell of an $8\\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows suc... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e613a583-1a0f-46a5-a2c7-16253beb4397"
} | [
{
"content": "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\nEvery cell of an $8\\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows suc... |
In how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls? | 393 | math_dapo | [
{
"content": "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\nIn how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2... | MATH | {
"ground_truth": "393",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "36999dfa-b3a9-447b-9d67-848b54a48716"
} | [
{
"content": "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\nIn how many ways can $7$ red and $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2... |
Triangle $ABC$ is a right triangle. If the measure of angle $PAB$ is $x^\circ$ and the measure of angle $ACB$ is expressed in the form $(Mx+N)^\circ$ with $M=1$, what is the value of $M+N$?
[asy]
draw((-10,0)--(20,0),linewidth(1),Arrows);
draw((0,0)--(10,10/sqrt(3))--(10+10/3,0),linewidth(1));
draw((10,10/sqrt(3))+di... | -89 | math_dapo | [
{
"content": "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\nTriangle $ABC$ is a right triangle. If the measure of angle $PAB$ is $x^\\circ$ and the measure of angle $ACB$ is expr... | MATH | {
"ground_truth": "-89",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "176ba670-6e86-41ad-83b4-bc7baf39ad90"
} | [
{
"content": "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\nTriangle $ABC$ is a right triangle. If the measure of angle $PAB$ is $x^\\circ$ and the measure of angle $ACB$ is expr... |
The area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inches. What is the perimeter of the non-shaded region?
[asy]size(101);
filldraw(((0,0)--(0,8)--(10,8)--(10,-2)--(6,-2)--(6,0)--cycle^^(2.5,3)--(2.5,5)--(7.5,5)--(7.5,3)--cycle),gray(.6)+fillrule(1),linewid... | 14 | math_dapo | [
{
"content": "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\nThe area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inche... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b21001da-4c4b-4f4d-8dfb-717d7ab6a486"
} | [
{
"content": "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\nThe area of the shaded region is 78 square inches. All angles are right angles and all measurements are given in inche... |
Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his s... | 751 | math_dapo | [
{
"content": "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\nPrinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an intege... | MATH | {
"ground_truth": "751",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "46b7065f-639b-488f-93ca-510e3191b98f"
} | [
{
"content": "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\nPrinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an intege... |
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