prompt stringlengths 19 3.19k | solution stringlengths 1 23 | 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 \( a\uparrow\uparrow b = a^{a^{\cdots^{a}}} \), where there are \( b \) instances of \( a \). The operation \( a\uparrow\uparrow b \) is defined recursively as follows:
\[ a\uparrow\uparrow b = \begin{cases} a & \text{if } b=1 \\ a^{a\uparrow\uparrow (b-1)} & \text{if } b\ge2 \end{cases} \]
Determine the remai... | 27 | 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\\uparrow\\uparrow b = a^{a^{\\cdots^{a}}} \\), where there are \\( b \\) instances of \\( a \\). The operati... | MATH | {
"ground_truth": "27",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "74819e2f-feec-41c5-954f-83cc1f3244e4"
} | [
{
"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\\uparrow\\uparrow b = a^{a^{\\cdots^{a}}} \\), where there are \\( b \\) instances of \\( a \\). The operati... |
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible... | 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\nLet $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\\angle BAC$ intersect line $BC$ at $D$. If... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6eb0a35b-bc1f-4bc5-b842-0572fdc0f5e7"
} | [
{
"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 such that $AB = 7$, and let the angle bisector of $\\angle BAC$ intersect line $BC$ at $D$. If... |
Let Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, "If I give you some marbles then you will have twice as many marbles as I will have." Birbal says to Akbar, "If I give you some marbles then you will have thrice as many marbles as I will have." What is the minimum possible value of $n... | 12 | 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 Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, \"If I give you some marbles then... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8b313ff9-0a61-4766-b733-9e0c19d2e5cc"
} | [
{
"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 Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, \"If I give you some marbles then... |
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | 20 | 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 equilateral $\\triangle ABC$ let points $D$ and $E$ trisect $\\overline{BC}$. Then $\\sin(\\angle DAE)$ can be expr... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "438fb207-70d5-4f45-9e28-3119a9971291"
} | [
{
"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 equilateral $\\triangle ABC$ let points $D$ and $E$ trisect $\\overline{BC}$. Then $\\sin(\\angle DAE)$ can be expr... |
Let $f(x) = \sin(\sin(x))$. Evaluate:
\[
\lim_{h \to 0} \frac{f(x+h) - f(h)}{x}
\] at $x = \pi$. | 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 $f(x) = \\sin(\\sin(x))$. Evaluate:\n\\[\n\\lim_{h \\to 0} \\frac{f(x+h) - f(h)}{x}\n\\] at $x = \\pi$.\n\nRemembe... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d0b05edf-c496-4b37-8fd2-0e2cf984ad6f"
} | [
{
"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 $f(x) = \\sin(\\sin(x))$. Evaluate:\n\\[\n\\lim_{h \\to 0} \\frac{f(x+h) - f(h)}{x}\n\\] at $x = \\pi$.\n\nRemembe... |
Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\frac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability... | 743 | 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\nKeith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the p... | MATH | {
"ground_truth": "743",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb5bd29d-1f52-43b6-8948-ada15b3095d8"
} | [
{
"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\nKeith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the p... |
Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. | 247 | 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 $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\\overline{ABC}_7$, and in... | MATH | {
"ground_truth": "247",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "14b80dbe-c7b3-48d1-a069-3c67c762ed3f"
} | [
{
"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 $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\\overline{ABC}_7$, and in... |
Rectangle $ABCD$ has a perimeter of $178$ and an area of $1848$. What is the length of the diagonal of the rectangle? | 65 | 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\nRectangle $ABCD$ has a perimeter of $178$ and an area of $1848$. What is the length of the diagonal of the rectangle?\... | MATH | {
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1636ec81-fce1-4700-b733-4d5e77df8011"
} | [
{
"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\nRectangle $ABCD$ has a perimeter of $178$ and an area of $1848$. What is the length of the diagonal of the rectangle?\... |
Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$. | 4306 | 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 $x, y$ be positive real numbers. If \\[129-x^2=195-y^2=xy,\\] then $x = \\frac{m}{n}$ for relatively prime positiv... | MATH | {
"ground_truth": "4306",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e9ca2e43-fc12-4f4d-8bd2-2bcd9633aab4"
} | [
{
"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 $x, y$ be positive real numbers. If \\[129-x^2=195-y^2=xy,\\] then $x = \\frac{m}{n}$ for relatively prime positiv... |
There are three bags:
1. Bag A contains 3 green candies and 1 red candy.
2. Bag B contains 2 green candies and 2 red candies.
3. Bag C contains 1 green candy and 3 red candies.
A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy.
- If the first candy is green, t... | 217 | 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 three bags:\n1. Bag A contains 3 green candies and 1 red candy.\n2. Bag B contains 2 green candies and 2 red... | MATH | {
"ground_truth": "217",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a00d95da-ed53-4885-b162-9268b2817515"
} | [
{
"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 three bags:\n1. Bag A contains 3 green candies and 1 red candy.\n2. Bag B contains 2 green candies and 2 red... |
Three points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances between the points are less than the radius of the circle. The probability is expressed in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\). | 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\nThree points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a6545fbc-30c4-4dff-9536-b7abc15ef302"
} | [
{
"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\nThree points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances... |
Jesse has ten squares, labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors? | 2304 | 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\nJesse has ten squares, labeled $1, 2, \\dots, 10$. In how many ways can he color each square either red, green, yellow... | MATH | {
"ground_truth": "2304",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ab2595c8-eb8e-4af5-b728-df23c236f7ca"
} | [
{
"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\nJesse has ten squares, labeled $1, 2, \\dots, 10$. In how many ways can he color each square either red, green, yellow... |
Line $\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are the orthogonal projections of points $B$ and $C$ onto line $\ell$, respectively. Given that $DE = 1$ and $2BD = CE$, determine the area of triangle $ABC$ expressed as $m\sqrt{n}$, where $m$ and $n$ are p... | 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\nLine $\\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are ... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f22c9aa4-1fa9-44dc-9a58-9e88c477ba8a"
} | [
{
"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\nLine $\\ell$ passes through point $A$ and into the interior of the equilateral triangle $ABC$. Points $D$ and $E$ are ... |
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime ... | 200 | 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 the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\l... | MATH | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "17051c97-34b2-43e8-8f67-80ab35c16c74"
} | [
{
"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 the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\l... |
Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation has more $1$ 's than $0$ 's. Find the remainder when $N$ is divided by $1000$ . | 155 | 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 $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation ha... | MATH | {
"ground_truth": "155",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1c9d8428-3f0a-4a70-92cb-54a01835ffbc"
} | [
{
"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 $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation ha... |
In the staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight congruent sides marked with a tick mark have length 1 foot. If the region has area 53 square feet, what is the number of feet in the perimeter of the region? [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(... | 32 | 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 staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight c... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a3c2df9a-f2cb-42f7-95a0-58c21bdf665b"
} | [
{
"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 staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight c... |
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