prompt stringlengths 19 3.19k | solution stringlengths 1 23 | data_source stringclasses 1
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|---|---|---|---|---|---|---|---|
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}$. Compute the sum:
\[ \sum_{i=1}^{1000} g(i). \] | 8000 | 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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}... | MATH | {
"ground_truth": "8000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "141399c6-e07a-49ac-aefd-8c267b621c6a"
} | [
{
"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 nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}... |
What is the smallest positive integer $n$ such that, out of the $n$ unit fractions $\frac{1}{k}$ where $1 \le k \le n$, exactly half of the fractions give a terminating decimal? | 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\nWhat is the smallest positive integer $n$ such that, out of the $n$ unit fractions $\\frac{1}{k}$ where $1 \\le k \\le... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c8842d88-54dd-45ec-8945-f5c8836f5722"
} | [
{
"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 smallest positive integer $n$ such that, out of the $n$ unit fractions $\\frac{1}{k}$ where $1 \\le k \\le... |
After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they mu... | 982 | 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\nAfter moving his sticky toy spider one morning, Tony heads outside to play \"pirates\" with his pal Nick, who lives a ... | MATH | {
"ground_truth": "982",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ba8824ed-47eb-4743-bdbe-86090e2e6672"
} | [
{
"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\nAfter moving his sticky toy spider one morning, Tony heads outside to play \"pirates\" with his pal Nick, who lives a ... |
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$, which is the greatest integer not exceeding... | 157 | 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 $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\\ell$ passing through $B$, we color ... | MATH | {
"ground_truth": "157",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2c557f8d-1f97-4480-a4cc-8e37b371b849"
} | [
{
"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 $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\\ell$ passing through $B$, we color ... |
Find the sum of all possible $n$ such that $n$ is a positive integer and there exist real numbers $a, b, c$ such that for every integer $m$, the quantity \( \frac{2013m^3 + am^2 + bm + c}{n} \) is an integer. | 2976 | 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 sum of all possible $n$ such that $n$ is a positive integer and there exist real numbers $a, b, c$ such that ... | MATH | {
"ground_truth": "2976",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9c982f7d-fe9e-4c23-b502-58c98a0b76bf"
} | [
{
"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 sum of all possible $n$ such that $n$ is a positive integer and there exist real numbers $a, b, c$ such that ... |
All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon? | 161 | 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\nAll the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals co... | MATH | {
"ground_truth": "161",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "23e52553-5dfe-4de8-b0a4-7367b86e5ba8"
} | [
{
"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\nAll the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals co... |
A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14... | -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\nA portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance betwee... | MATH | {
"ground_truth": "-8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0906a189-c3fd-46ae-a7ab-afaa55b131b0"
} | [
{
"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 portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance betwee... |
Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are i... | 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\nFind the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length ... | MATH | {
"ground_truth": "15",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8569d53f-c503-4563-96e9-b43ae0196c17"
} | [
{
"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 largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length ... |
Given that the point $(8,8)$ is on the graph of $y=\frac 14f\left(\frac 12x\right)$, there is one point that must be on the graph of $y=f(x)$. What is the sum of coordinates of that point? | 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\nGiven that the point $(8,8)$ is on the graph of $y=\\frac 14f\\left(\\frac 12x\\right)$, there is one point that must ... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "35267123-db81-478c-9fbe-ab57437d73f1"
} | [
{
"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 that the point $(8,8)$ is on the graph of $y=\\frac 14f\\left(\\frac 12x\\right)$, there is one point that must ... |
A village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is located outside the village, precisely $16 \, \mathrm{m}$ north of the north gate. This tree is visible on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. Determine the diameter, i... | 48 | 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 village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree i... | MATH | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6278bfd6-6e56-4e83-8003-d2d11799ebf1"
} | [
{
"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 village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree i... |
Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box?
```
\begin{asy}
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));
draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1)... | 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\nFill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is t... | MATH | {
"ground_truth": "118",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c3a9fde1-a388-46f3-a801-b164ccad79ab"
} | [
{
"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\nFill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is t... |
Given positive integers $x$ and $y$ such that $x\neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{18}$, what is the smallest possible value for $x + y$? | 75 | 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 positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{18}$, what is th... | MATH | {
"ground_truth": "75",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1afae752-f613-4cf9-8aaa-482af576d145"
} | [
{
"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 positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{18}$, what is th... |
7. Let $r_1, r_2, r_3$ be the roots of the polynomial equation $x^3 - 2x^2 + 4x + 10 = 0$. Compute the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$. | 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\n7. Let $r_1, r_2, r_3$ be the roots of the polynomial equation $x^3 - 2x^2 + 4x + 10 = 0$. Compute the value of $(r_1 ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c90021ec-3d77-45a7-a752-0df26951b7b3"
} | [
{
"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\n7. Let $r_1, r_2, r_3$ be the roots of the polynomial equation $x^3 - 2x^2 + 4x + 10 = 0$. Compute the value of $(r_1 ... |
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ a... | 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\nEvaluate$$\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n}\\left(\\left[\\frac{2 n}{k}\\right]-2\\left[\\f... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fb9ed406-5c7a-4982-9dd2-13306b208733"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate$$\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n}\\left(\\left[\\frac{2 n}{k}\\right]-2\\left[\\f... |
Compute the last two digits of $9^{2020} + 9^{2020^2} + \cdots + 9^{2020^{2020}}$. | 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\nCompute the last two digits of $9^{2020} + 9^{2020^2} + \\cdots + 9^{2020^{2020}}$.\n\nRemember to put your answer on ... | MATH | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "03d45551-828b-4ce7-99bb-8c0977ad8dcc"
} | [
{
"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 last two digits of $9^{2020} + 9^{2020^2} + \\cdots + 9^{2020^{2020}}$.\n\nRemember to put your answer on ... |
In a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some air company. Find the minimal possible number of air companies required if direct flights between any three cities are operated by three different air companies. | 2015 | 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 country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by... | MATH | {
"ground_truth": "2015",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c7a8a061-ce41-4200-92b6-78af4fbbfb34"
} | [
{
"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 country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by... |
The degree-$6$ polynomial $f$ satisfies the following conditions:
- $f(7) - f(1) = 1$,
- $f(8) - f(2) = 16$,
- $f(9) - f(3) = 81$,
- $f(10) - f(4) = 256$,
- $f(11) - f(5) = 625$.
Compute $f(15) - f(-3)$. | 6723 | 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 degree-$6$ polynomial $f$ satisfies the following conditions: \n- $f(7) - f(1) = 1$, \n- $f(8) - f(2) = 16$, \n... | MATH | {
"ground_truth": "6723",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1812dffc-9930-4ce5-9b15-996f661b2ae0"
} | [
{
"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 degree-$6$ polynomial $f$ satisfies the following conditions: \n- $f(7) - f(1) = 1$, \n- $f(8) - f(2) = 16$, \n... |
How many sets of two or more consecutive positive integers have a sum of $15$? Provide your answer as an 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\nHow many sets of two or more consecutive positive integers have a sum of $15$? Provide your answer as an integer.\n\nR... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "271069be-e62e-4923-a60f-cf070c0fb139"
} | [
{
"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 sets of two or more consecutive positive integers have a sum of $15$? Provide your answer as an integer.\n\nR... |
Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that $74$ are on the football team; $26$ are on the baseball team; $17$ are on both the football and lacrosse teams; $18$ are on both the baseball and footb... | 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\nVernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: fo... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8f264a19-7e74-4b7b-9786-a0721b37d569"
} | [
{
"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\nVernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: fo... |
Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$. | -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 sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.\n\nRemember to put your answer on its own lin... | MATH | {
"ground_truth": "-7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "60fbd57c-2df1-4d85-b0e0-c50413d5895b"
} | [
{
"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 sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.\n\nRemember to put your answer on its own lin... |
How many even integers between $4000$ and $7000$ have four different digits? | 728 | 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 even integers between $4000$ and $7000$ have four different digits?\n\nRemember to put your answer on its own... | MATH | {
"ground_truth": "728",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9b39b49c-da7a-4393-bd09-acfc20e1b955"
} | [
{
"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 even integers between $4000$ and $7000$ have four different digits?\n\nRemember to put your answer on its own... |
If a planet of radius $R$ spins with an angular velocity $\omega$ about an axis through the North Pole, find the ratio of the normal force experienced by a person at the equator to that experienced by a person at the North Pole. Assume a constant gravitational field $g$ and that both people are stationary relative to t... | 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 a planet of radius $R$ spins with an angular velocity $\\omega$ about an axis through the North Pole, find the rati... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a8426794-1665-428e-be8d-ffecb3b1d3e9"
} | [
{
"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 a planet of radius $R$ spins with an angular velocity $\\omega$ about an axis through the North Pole, find the rati... |
Determine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all values of $a$ and $p$ for the pairs that satisfy the condition. | 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\nDetermine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5ac1bab6-fba8-4592-a908-c2c13fedaaa0"
} | [
{
"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 all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please... |
What is the largest positive integer \( n \) satisfying the following inequality:
\[ n^{2006} < 7^{2007} \] | 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\nWhat is the largest positive integer \\( n \\) satisfying the following inequality:\n\\[ n^{2006} < 7^{2007} \\]\n\nRe... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0ffce7d4-83e1-4487-a735-4ed26d9c5087"
} | [
{
"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 largest positive integer \\( n \\) satisfying the following inequality:\n\\[ n^{2006} < 7^{2007} \\]\n\nRe... |
If $x \equiv 4 \pmod{19}$ and $y \equiv 7 \pmod{19}$, then find the remainder when $(x + 1)^2 (y + 5)^3$ is divided by 19. | 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\nIf $x \\equiv 4 \\pmod{19}$ and $y \\equiv 7 \\pmod{19}$, then find the remainder when $(x + 1)^2 (y + 5)^3$ is divide... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e2e4e480-2328-474a-b05b-b5c9f4950a87"
} | [
{
"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 $x \\equiv 4 \\pmod{19}$ and $y \\equiv 7 \\pmod{19}$, then find the remainder when $(x + 1)^2 (y + 5)^3$ is divide... |
Find all solutions to the equation $\sqrt{3x+6}=x+2$. If there are multiple solutions, order them from least to greatest, separated by comma(s). | 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\nFind all solutions to the equation $\\sqrt{3x+6}=x+2$. If there are multiple solutions, order them from least to grea... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b3fab2c1-2651-47c1-93f9-0b7e71ac43c5"
} | [
{
"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 solutions to the equation $\\sqrt{3x+6}=x+2$. If there are multiple solutions, order them from least to grea... |
Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio o... | 2449 | 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\nFour congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four... | MATH | {
"ground_truth": "2449",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "81d9ce13-582d-4937-b4d5-3d80c2c466c5"
} | [
{
"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\nFour congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four... |
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are... | 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\nTwo mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times ... | MATH | {
"ground_truth": "87",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "65d29b26-5793-4428-a1ea-b6e476e83cc9"
} | [
{
"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 mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times ... |
Let $x$ and $y$ be positive real numbers that satisfy $(\log x)^2 + (\log y)^2 = \log(x^2) + \log(y^2)$. Compute the maximum possible value of $(\log(xy))^2$. | 16 | 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 $x$ and $y$ be positive real numbers that satisfy $(\\log x)^2 + (\\log y)^2 = \\log(x^2) + \\log(y^2)$. Compute t... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "04edd971-ea9e-4b17-b336-53043e8f30fb"
} | [
{
"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 $x$ and $y$ be positive real numbers that satisfy $(\\log x)^2 + (\\log y)^2 = \\log(x^2) + \\log(y^2)$. Compute t... |
Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\]
for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$.
(Note: an invertible function is one that has an inverse). | 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 $f$ be an invertible function defined on the complex numbers such that \\[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + ... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "db55224a-7444-46e6-8e09-49221c53e854"
} | [
{
"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 $f$ be an invertible function defined on the complex numbers such that \\[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + ... |
Given the following system of equations:
\[ a+4b+9c+16d+25e+36f+49g=1 \]
\[ 4a+9b+16c+25d+36e+49f+64g=12 \]
\[ 9a+16b+25c+36d+49e+64f+81g=123 \]
Determine the value of \( 16a+25b+36c+49d+64e+81f+100g \). | 334 | 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 the following system of equations: \n\\[ a+4b+9c+16d+25e+36f+49g=1 \\] \n\\[ 4a+9b+16c+25d+36e+49f+64g=12 \\] ... | MATH | {
"ground_truth": "334",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7a5316bb-a39b-406c-bbcd-9f9abea31101"
} | [
{
"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 the following system of equations: \n\\[ a+4b+9c+16d+25e+36f+49g=1 \\] \n\\[ 4a+9b+16c+25d+36e+49f+64g=12 \\] ... |
Find the radius of the inscribed circle of a triangle with sides of length \(13\), \(30\), and \(37\). Provide your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Calculate the value of \(k + m\). | 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 radius of the inscribed circle of a triangle with sides of length \\(13\\), \\(30\\), and \\(37\\). Provide y... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d3cc2308-e321-4715-b22e-901826c8ae87"
} | [
{
"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 radius of the inscribed circle of a triangle with sides of length \\(13\\), \\(30\\), and \\(37\\). Provide y... |
Arthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he walk? | 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\nArthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he wal... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "719ad1fa-8663-4efe-8e3d-a9761d3c3564"
} | [
{
"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\nArthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he wal... |
The function $f$ satisfies the functional equation \[ f(x) + f(y) = f(x + y) - xy - 1 \] for every pair $x, y$ of real numbers. If $f(1) = 1$, find the number of integers $n \neq 1$ for which $f(n) = n$. | 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 function $f$ satisfies the functional equation \\[ f(x) + f(y) = f(x + y) - xy - 1 \\] for every pair $x, y$ of re... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7ca5207a-bdc6-48d7-a525-532621c09bb5"
} | [
{
"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 $f$ satisfies the functional equation \\[ f(x) + f(y) = f(x + y) - xy - 1 \\] for every pair $x, y$ of re... |
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if... | 1507509 | 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 large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For exa... | MATH | {
"ground_truth": "1507509",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d63f9c3c-aacc-45e8-98d9-5d03d8415a90"
} | [
{
"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 large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For exa... |
Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal. | 1010 | 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 maximum integral value of $k$ such that $0 \\le k \\le 2019$ and $|e^{2\\pi i \\frac{k}{2019}} - 1|$ is maxim... | MATH | {
"ground_truth": "1010",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "de93ac99-011a-4608-954c-4efc6119a73f"
} | [
{
"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 maximum integral value of $k$ such that $0 \\le k \\le 2019$ and $|e^{2\\pi i \\frac{k}{2019}} - 1|$ is maxim... |
$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFBA$?
[asy]
size(4cm,4cm);
for(int i=0; i < 4; ++i){
for(int k=0; k < 5; ++k){
draw((0,i)--(4,i));
draw((k,0)--(k,3));
} }
draw((0,0)--(1,3));
draw((3,3)--(4,0));
label("$A$",(0,0),SW);
label("$B$",(... | 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 $Answer is the answer to the problem.\n\n$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFB... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cafffbac-75ae-4e8b-b65e-ea5586e64cb5"
} | [
{
"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 $Answer is the answer to the problem.\n\n$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFB... |
Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, indiv... | 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\nDexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intru... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "29806b03-feb0-4233-90cf-297ce2feda67"
} | [
{
"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\nDexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intru... |
Determine the number of ways to arrange the letters of the word GAMMAS. | 180 | 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 number of ways to arrange the letters of the word GAMMAS.\n\nRemember to put your answer on its own line... | MATH | {
"ground_truth": "180",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e7d483e5-599c-488d-b128-4416994e7267"
} | [
{
"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 number of ways to arrange the letters of the word GAMMAS.\n\nRemember to put your answer on its own line... |
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (the angle between the two planes). ... | 450 | 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\n Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and... | MATH | {
"ground_truth": "450",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bcca1981-3344-4d47-9503-4a3d815b9307"
} | [
{
"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\n Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and... |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > 0.$ Determine $f(2009).$ | 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\nLet $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ an... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "84e1e38d-f605-48d0-a7e1-a50ddfdd3a8f"
} | [
{
"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 $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ an... |
In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then what is the area of $\triangle AEF$? | 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\nIn triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the a... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4dff749b-5f2d-485a-8c55-ad0fd56485bf"
} | [
{
"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 $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the a... |
A sequence of positive integers is constructed by listing the first 4, then skipping one, listing the next 5, skipping 2, listing 6, skipping 3, and, on the $n$th iteration, listing $n + 3$ and skipping $n$. The sequence begins 1, 2, 3, 4, 6, 7, 8, 9, 10, 13. What is the 500,000th number in the sequence? | 996506 | 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 sequence of positive integers is constructed by listing the first 4, then skipping one, listing the next 5, skipping... | MATH | {
"ground_truth": "996506",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c6e81d28-e632-4b79-ba45-e527a5ecb8e8"
} | [
{
"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 sequence of positive integers is constructed by listing the first 4, then skipping one, listing the next 5, skipping... |
What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? | 99997 | 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 largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5?\... | MATH | {
"ground_truth": "99997",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "54eceefe-6a25-4452-b981-04c7dece9c91"
} | [
{
"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 largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5?\... |
Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$. | 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\nLet $a$ be a real number greater than $1$ such that $\\frac{20a}{a^2+1} = \\sqrt{2}$. Find $\\frac{14a}{a^2 - 1}$.\n\n... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "81480b4b-9805-4a7f-be1e-c000c941d9f7"
} | [
{
"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 $a$ be a real number greater than $1$ such that $\\frac{20a}{a^2+1} = \\sqrt{2}$. Find $\\frac{14a}{a^2 - 1}$.\n\n... |
A frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13$. A move sequence... | 169 | 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 frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the fr... | MATH | {
"ground_truth": "169",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bd45e6d9-5191-4d01-98f2-cede91a5162f"
} | [
{
"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 frog is placed at the origin on the number line , and moves according to the following rule: in a given move, the fr... |
Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_1+a_2+ \cdots +a_{100}=100 \quad \text{and} \quad
a_{101}+a_{102}+ \cdots + a_{200}=200.
$$What is the value of $a_2 - a_1$? 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. | 101 | 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 $\\{a_n\\}$ is an arithmetic sequence with $$\na_1+a_2+ \\cdots +a_{100}=100 \\quad \\text{and} \\quad\n... | MATH | {
"ground_truth": "101",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8c778472-087a-4811-ada0-adb829302256"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that $\\{a_n\\}$ is an arithmetic sequence with $$\na_1+a_2+ \\cdots +a_{100}=100 \\quad \\text{and} \\quad\n... |
At the beginning of the school year, $50\%$ of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end o... | 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\nAt the beginning of the school year, $50\\%$ of all students in Mr. Well's class answered \"Yes\" to the question \"Do... | MATH | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9f00bb3a-661d-48ed-84de-caeda2ca7231"
} | [
{
"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\nAt the beginning of the school year, $50\\%$ of all students in Mr. Well's class answered \"Yes\" to the question \"Do... |
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$? | 5 | 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 pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \\frac {3}{2008} \\equal{} \\frac 1m... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a59b7a6a-7b3e-4ff0-95c7-fa91824c72b1"
} | [
{
"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 pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \\frac {3}{2008} \\equal{} \\frac 1m... |
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ an... | 137 | 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\n$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any positio... | MATH | {
"ground_truth": "137",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e46e0ed8-dde9-4b2b-b569-83e5ae7a3506"
} | [
{
"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\n$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any positio... |
Consider the sequence defined by $a_k =\dfrac{1}{k^2+k}$ for $k\geq 1$ . Given that $a_m+a_{m+1}+\cdots+a_{n-1}=\dfrac{1}{29}$ , for positive integers $m$ and $n$ with $m<n$ , find $m+n$ . | 840 | 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\nConsider the sequence defined by $a_k =\\dfrac{1}{k^2+k}$ for $k\\geq 1$ . Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=\\d... | MATH | {
"ground_truth": "840",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "96c05a4a-25b2-47cd-befb-8dc1d3a52baf"
} | [
{
"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\nConsider the sequence defined by $a_k =\\dfrac{1}{k^2+k}$ for $k\\geq 1$ . Given that $a_m+a_{m+1}+\\cdots+a_{n-1}=\\d... |
Circle $\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\sqrt{3}$. Circle $\Omega$ is centered on the $x$-axis, passes through the point $A = (6066, 0)$, and intersects $\Gamma$ orthogonally at the point $P = (x, y)$ with $y > 0$. If the length of the minor arc $AP$ on $\Omega$ can be expressed as $\frac{... | 1349 | 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\nCircle $\\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\\sqrt{3}$. Circle $\\Omega$ is centered on the... | MATH | {
"ground_truth": "1349",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "da1b9ee8-0a99-433f-9c55-e9f4c6efb62e"
} | [
{
"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\nCircle $\\Gamma$ is centered at $(0, 0)$ in the plane with radius $2022\\sqrt{3}$. Circle $\\Omega$ is centered on the... |
Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km... | 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\nMinnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny ride... | MATH | {
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "34d12d8d-4f52-480c-bea8-c0fce73e52ee"
} | [
{
"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\nMinnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny ride... |
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? | 1012 | 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\nSolomiya wrote the numbers $1, 2, \\ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from ... | MATH | {
"ground_truth": "1012",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a51c8897-4751-4629-8246-303cada01dee"
} | [
{
"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\nSolomiya wrote the numbers $1, 2, \\ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from ... |
Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$. | -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\nSuppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all pos... | MATH | {
"ground_truth": "-90",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "31dcbcb2-3e72-4e8c-b021-5e8007e5d7c8"
} | [
{
"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 $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all pos... |
Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$. | 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\nLet $z$ be a complex number satisfying $(z+\\frac{1}{z})(z+\\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\\frac{2}{z^{100}}+... | MATH | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "db6da960-1395-4a54-8cfc-b586db851a54"
} | [
{
"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 $z$ be a complex number satisfying $(z+\\frac{1}{z})(z+\\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\\frac{2}{z^{100}}+... |
Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$. | 37 | 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\nGary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers.... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b8b62bad-c062-4d1d-9c76-053440ca16a5"
} | [
{
"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\nGary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers.... |
What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\). | 142 | 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 value of \\[3+\\frac{1}{3+\\frac{1}{3+\\frac{1}{3}}}?\\] The original answer is in the form \\(\\frac{k}{m... | MATH | {
"ground_truth": "142",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "855eec71-0e4b-48f3-a147-d7aa041500bb"
} | [
{
"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 value of \\[3+\\frac{1}{3+\\frac{1}{3+\\frac{1}{3}}}?\\] The original answer is in the form \\(\\frac{k}{m... |
Let $ABC$ be a triangle with $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpendicular bisector of $AC$, and the line through $I$ perpendicular to $AB$ intersect at a common point, then the length $AB$ can be written as $m + \sqrt{n}$ for positive integers $m$ and $n$. Wh... | 460 | 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 $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, t... | MATH | {
"ground_truth": "460",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "78ed26a7-9cc4-4d51-ad36-3ade94043dfc"
} | [
{
"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 $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, t... |
What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 37\cdot 2x + 37^2$ is a multiple of 47? | 5 | 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 least positive integer value of $x$ such that $(2x)^2 + 2\\cdot 37\\cdot 2x + 37^2$ is a multiple of 47?\n... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cc9ec722-5333-46c2-8986-ac020e61d259"
} | [
{
"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 least positive integer value of $x$ such that $(2x)^2 + 2\\cdot 37\\cdot 2x + 37^2$ is a multiple of 47?\n... |
Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter such that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of th... | 47 | 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 $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same dia... | MATH | {
"ground_truth": "47",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3b0aa3f3-39f6-43c4-addf-5b997353a4cf"
} | [
{
"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 $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same dia... |
Square $ABCD$ has sides of length 4, and $M$ is the midpoint of $\overline{CD}$. A circle with radius 2 and center $M$ intersects a circle with radius 4 and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$? Express your answer as a common fraction.
[asy]
pair A,B,C,D,M,P;
D=(0,0);
C=(... | 21 | 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\nSquare $ABCD$ has sides of length 4, and $M$ is the midpoint of $\\overline{CD}$. A circle with radius 2 and center $M... | MATH | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49264729-ace2-435f-ad20-d12514cc67c9"
} | [
{
"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\nSquare $ABCD$ has sides of length 4, and $M$ is the midpoint of $\\overline{CD}$. A circle with radius 2 and center $M... |
You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $1$ m/s. Thereafter, when exactly $1$ second has passed from when Al last changed its speed, Al’s speed instantaneously become... | 210 | 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\nYou are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m... | MATH | {
"ground_truth": "210",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4db14b96-b7f5-4c21-983c-17c2015de3e7"
} | [
{
"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\nYou are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m... |
A subset $B$ of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$? | 62 | 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 subset $B$ of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of $B$ sum to ... | MATH | {
"ground_truth": "62",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c1496f86-02ad-42fe-beaa-618dc5e19b1f"
} | [
{
"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 subset $B$ of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of $B$ sum to ... |
Find all prime numbers $p$ such that the expression $p^2 - p + 1$ is a perfect cube. | 19 | 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 the expression $p^2 - p + 1$ is a perfect cube.\n\nRemember to put your answer on... | MATH | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0a788f82-3939-4145-be23-d08d9c574210"
} | [
{
"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 the expression $p^2 - p + 1$ is a perfect cube.\n\nRemember to put your answer on... |
Starting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365$. Find $y + d$. | 317 | 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\nStarting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d... | MATH | {
"ground_truth": "317",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d15a93e9-3943-4cb8-b65d-4bd34054e0bf"
} | [
{
"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\nStarting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d... |
For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice? Provide your answer as a fraction in the form \(\frac{k}{m}\) and calculate the value of \(k + m\). | 71 | 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 a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are i... | MATH | {
"ground_truth": "71",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7d27f8e7-a868-4630-add5-c20b826b0c42"
} | [
{
"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 a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are i... |
Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$, find $a+b+... | 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\nLet $P_{1}: y=x^{2}+\\frac{101}{100}$ and $P_{2}: x=y^{2}+\\frac{45}{4}$ be two parabolas in the Cartesian plane. Let ... | MATH | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "16dc4a27-b0e5-4091-a0b0-59a15e845892"
} | [
{
"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 $P_{1}: y=x^{2}+\\frac{101}{100}$ and $P_{2}: x=y^{2}+\\frac{45}{4}$ be two parabolas in the Cartesian plane. Let ... |
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$, as shown. What is the perimeter of this region? | 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\nA region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\\frac{2}{\\pi}$, as... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3bff2879-4732-4643-812b-f32187d8ae1c"
} | [
{
"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 region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\\frac{2}{\\pi}$, as... |
A stadium staircase with 150 steps is numbered from 1 to 150. Jan starts on step 130 and steps down to step number 127, then to step number 124, and continues downward to step 13 stepping only on every third step. Jen starts on step number 3 and steps up to step number 7, then to step number 11, and continues upward to... | 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 stadium staircase with 150 steps is numbered from 1 to 150. Jan starts on step 130 and steps down to step number 127... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2ff86ba9-17f4-449d-9eec-81ca5755fa2d"
} | [
{
"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 stadium staircase with 150 steps is numbered from 1 to 150. Jan starts on step 130 and steps down to step number 127... |
Let $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\angle ABC = 30^\circ$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees. | 80 | 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 $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\\angle ABC = 30^\\circ$ ... | MATH | {
"ground_truth": "80",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "90089b5b-44d3-44bf-bf05-fe520bee9c73"
} | [
{
"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 $AB$ be a chord of a circle with center $O$. Let $C$ be a point on the circle such that $\\angle ABC = 30^\\circ$ ... |
Given $\cos (\alpha + \beta) + \sin (\alpha - \beta) = 0$ and $\tan \beta = \frac{1}{2000},$ find $\tan \alpha.$ | -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\nGiven $\\cos (\\alpha + \\beta) + \\sin (\\alpha - \\beta) = 0$ and $\\tan \\beta = \\frac{1}{2000},$ find $\\tan \\al... | MATH | {
"ground_truth": "-1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "37bd6d19-849d-45a8-b193-8d84036b9a41"
} | [
{
"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 $\\cos (\\alpha + \\beta) + \\sin (\\alpha - \\beta) = 0$ and $\\tan \\beta = \\frac{1}{2000},$ find $\\tan \\al... |
Michael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order.
Given a 7-digit number format, consider the number $987-1234$ where the first four digits $(9-8-7-1)$ des... | 2400 | 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\nMichael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, t... | MATH | {
"ground_truth": "2400",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8a2ef70a-74d6-4673-bd79-ae6b25376a01"
} | [
{
"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\nMichael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, t... |
How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$? | 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\nHow many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ h... | MATH | {
"ground_truth": "99",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "140a2781-070d-48d8-b4d0-7e6e888e1f3f"
} | [
{
"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 ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ h... |
Evaluate \( \int_0^{2008} \left(3x^2 - 8028x + 2007^2 + \frac{1}{2008}\right)\, dx \). | 2009 | 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\nEvaluate \\( \\int_0^{2008} \\left(3x^2 - 8028x + 2007^2 + \\frac{1}{2008}\\right)\\, dx \\).\n\nRemember to put your ... | MATH | {
"ground_truth": "2009",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "45a0468a-b7dc-46b2-a329-1791d6b5828b"
} | [
{
"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\nEvaluate \\( \\int_0^{2008} \\left(3x^2 - 8028x + 2007^2 + \\frac{1}{2008}\\right)\\, dx \\).\n\nRemember to put your ... |
Jack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the integers 1 through 6 (fair meaning the numbers appear with equal probability). Jack wins if a prime number appears when he rolls, while Jill wins if a number greater than 1 appears when she rolls. The game... | 16 | 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\nJack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the int... | MATH | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c74f7b5b-21b9-4e8c-9897-432f8b5d22ae"
} | [
{
"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\nJack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the int... |
Each of $\triangle PQR$ and $\triangle STU$ has an area of $1.$ In $\triangle PQR,$ $U,$ $W,$ and $V$ are the midpoints of the sides. In $\triangle STU,$ $R,$ $V,$ and $W$ are the midpoints of the sides. What is the area of parallelogram $UVRW?$ [asy]
draw((0,0)--(2,0)--(1,2)--cycle,linewidth(1));
draw((0,2)--(2,2)--(1... | 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\nEach of $\\triangle PQR$ and $\\triangle STU$ has an area of $1.$ In $\\triangle PQR,$ $U,$ $W,$ and $V$ are the midpo... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ed1f40cb-60f4-4815-9fa1-0d4d7f9d36f3"
} | [
{
"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\nEach of $\\triangle PQR$ and $\\triangle STU$ has an area of $1.$ In $\\triangle PQR,$ $U,$ $W,$ and $V$ are the midpo... |
The increasing list of five different integers $\{3,4,5,8,9\}$ has a sum of 29. How many increasing lists of five different single-digit positive integers have a sum of 33? | 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 increasing list of five different integers $\\{3,4,5,8,9\\}$ has a sum of 29. How many increasing lists of five d... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "86772a8e-c3a5-4d2e-9939-f576eff2dec3"
} | [
{
"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 increasing list of five different integers $\\{3,4,5,8,9\\}$ has a sum of 29. How many increasing lists of five d... |
The decimal representation of
$$\dfrac{1}{20^{20}}$$
consists of a string of zeros after the decimal point, followed by a $9$ and then several more digits. How many zeros are in that initial string of zeros after the decimal point? | 26 | 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 representation of \n$$\\dfrac{1}{20^{20}}$$ \nconsists of a string of zeros after the decimal point, follo... | MATH | {
"ground_truth": "26",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "723da097-c92c-493b-8354-df2cd375dd6e"
} | [
{
"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 representation of \n$$\\dfrac{1}{20^{20}}$$ \nconsists of a string of zeros after the decimal point, follo... |
Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions. | 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\nFind all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.\n\nRemember to ... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0406a53b-4202-4b22-bb9e-e2c224220fd3"
} | [
{
"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 values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.\n\nRemember to ... |
Let $f(n) = \frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c) = 2$. (Here, $f^5(x)$ means $f(f(f(f(f(x)))))$.) | 748 | 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 $f(n) = \\frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive int... | MATH | {
"ground_truth": "748",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "16e8cd82-b7ed-4384-aa25-58973a11a729"
} | [
{
"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 $f(n) = \\frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive int... |
Find the minimum natural number $n$ with the following property: between any collection of $n$ distinct natural numbers in the set $\{1,2, \dots,999\}$, it is possible to choose four different $a, b, c, d$ such that $a + 2b + 3c = d$. | 835 | 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 minimum natural number $n$ with the following property: between any collection of $n$ distinct natural number... | MATH | {
"ground_truth": "835",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "eaa5e7f8-5527-4b44-a55a-1b676c07e08b"
} | [
{
"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 minimum natural number $n$ with the following property: between any collection of $n$ distinct natural number... |
The first term of an arithmetic series of consecutive integers is \( k^2 + 1 \). Find the sum of \( 2k + 1 \) terms of this series. The original answer is in the form \( m \cdot k^3 + n \cdot (k + 1)^3 \). Please calculate the value of \( m + n \). | 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 first term of an arithmetic series of consecutive integers is \\( k^2 + 1 \\). Find the sum of \\( 2k + 1 \\) term... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "caec759b-d2aa-49ca-9098-4630c8a58173"
} | [
{
"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 first term of an arithmetic series of consecutive integers is \\( k^2 + 1 \\). Find the sum of \\( 2k + 1 \\) term... |
Find the sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original answer is in the form an(b + c), where a, b, and c are constants. Please give the value of a + b + c. | 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 sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original ans... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "46bcbc7e-c1d5-452c-9603-81b4c10d4614"
} | [
{
"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 sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original ans... |
Find the 3-digit positive integer that has the most divisors. | 840 | 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 3-digit positive integer that has the most divisors.\n\nRemember to put your answer on its own line after \"A... | MATH | {
"ground_truth": "840",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "69753844-8563-4c85-987b-298c5ec806f8"
} | [
{
"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 3-digit positive integer that has the most divisors.\n\nRemember to put your answer on its own line after \"A... |
What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? Fill in the blank with the correct number. | 48 | 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 smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of le... | MATH | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "23699c78-8730-46dc-b851-74eb606e1ed5"
} | [
{
"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 smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of le... |
Let $p(x)$ be a monic polynomial of degree 6 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and $p(6) = 6.$ Find $p(7).$ | 727 | 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 $p(x)$ be a monic polynomial of degree 6 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and... | MATH | {
"ground_truth": "727",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "18304b77-d112-4689-bff6-37db92889934"
} | [
{
"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 $p(x)$ be a monic polynomial of degree 6 such that $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ and... |
Juan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. As a result, Maria's number exceeds Juan's number by $14789$. Find the original number that Juan wrote. | 532 | 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\nJuan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. A... | MATH | {
"ground_truth": "532",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c6fc994b-3a37-404c-8896-632b131e340e"
} | [
{
"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\nJuan wrote a natural number. Maria then added a digit $1$ to the left and a digit $1$ to the right of Juan's number. A... |
Bill bought 13 notebooks, 26 pens, and 19 markers for $25. Paula bought 27 notebooks, 18 pens, and 31 markers for $31. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers? | 88 | 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\nBill bought 13 notebooks, 26 pens, and 19 markers for $25. Paula bought 27 notebooks, 18 pens, and 31 markers for $31.... | MATH | {
"ground_truth": "88",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e123a12d-5aae-4cd8-af79-c0b286698c3d"
} | [
{
"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\nBill bought 13 notebooks, 26 pens, and 19 markers for $25. Paula bought 27 notebooks, 18 pens, and 31 markers for $31.... |
Suppose $\cos R = \frac{4}{9}$ in the diagram below. What is $QS$?
[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(65),0);
R = (sqrt(65),-4);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,13));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$9$",(R+S)/2,SW);
[/asy]The answer is in the form k\sqrt{m... | 66 | 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 $\\cos R = \\frac{4}{9}$ in the diagram below. What is $QS$?\n\n\n\n[asy]\n\npair Q,R,S;\n\nS = (0,0);\n\nQ =... | MATH | {
"ground_truth": "66",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f2b97ccd-e716-429f-8e86-062b8d8b7489"
} | [
{
"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 $\\cos R = \\frac{4}{9}$ in the diagram below. What is $QS$?\n\n\n\n[asy]\n\npair Q,R,S;\n\nS = (0,0);\n\nQ =... |
Find the number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$. | 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\nFind the number of subsets of $\\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\}$ where the sum of the elements in the subset is... | MATH | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "094d5e36-2967-42f1-8d84-c46ba5613462"
} | [
{
"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 subsets of $\\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\}$ where the sum of the elements in the subset is... |
If \(a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + \frac{2}{5} = 0\) where \(a, b, c, d\) are real numbers, find the value of \(a\). The answer should be in the form \(\frac{k}{m}\), please provide the value of \(k + m\). | 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\nIf \\(a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + \\frac{2}{5} = 0\\) where \\(a, b, c, d\\) are real numbers, find the... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "43b6e388-a013-44bb-8d03-25e4e1db0dce"
} | [
{
"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 \\(a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + \\frac{2}{5} = 0\\) where \\(a, b, c, d\\) are real numbers, find the... |
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids? | 350 | 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 divided into two cuboids with integral side lengths by a straight cut. Afterwards, one... | MATH | {
"ground_truth": "350",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f5959f3f-361b-491d-809d-eb16538ef6a9"
} | [
{
"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 divided into two cuboids with integral side lengths by a straight cut. Afterwards, one... |
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is the length of... | 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\nTriangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\\angle{BAC}$ intersects $\\overline... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d8957d42-4f45-428b-9aea-993e8af35c8f"
} | [
{
"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 $AB = 11, BC=24$, and $CA = 20$. The bisector of $\\angle{BAC}$ intersects $\\overline... |
In triangle \(ABC\), side \(AC\) and the perpendicular bisector of \(BC\) meet at point \(D\), and \(BD\) bisects \(\angle ABC\). If \(AD=9\) and \(DC=7\), what is the area of triangle \(ABD\)? The original answer is in the form \(k\sqrt{m}\). Please provide the value of \(k + m\). | 19 | 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\\), side \\(AC\\) and the perpendicular bisector of \\(BC\\) meet at point \\(D\\), and \\(BD\\) bi... | MATH | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7b1570ee-d958-4c28-b15b-4c51e74171a5"
} | [
{
"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\\), side \\(AC\\) and the perpendicular bisector of \\(BC\\) meet at point \\(D\\), and \\(BD\\) bi... |
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)? | 61 | 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\nJeff writes down the two-digit base-$10$ prime $\\overline{ab_{10}}$. He realizes that if he misinterprets the number ... | MATH | {
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "53df866d-6ec3-49c3-9795-5ada8b5105c3"
} | [
{
"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\nJeff writes down the two-digit base-$10$ prime $\\overline{ab_{10}}$. He realizes that if he misinterprets the number ... |
Alice, Bob, and Carol play a game in which each of them chooses a real number between \(0\) and \(1.\) The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between \(0\) and \(1,\) ... | 37 | 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, Bob, and Carol play a game in which each of them chooses a real number between \\(0\\) and \\(1.\\) The winner ... | MATH | {
"ground_truth": "37",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "47075e30-0271-461a-9dba-c58496d6d61c"
} | [
{
"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, Bob, and Carol play a game in which each of them chooses a real number between \\(0\\) and \\(1.\\) The winner ... |
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the ... | 93 | 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\nAlfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person t... | MATH | {
"ground_truth": "93",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fb4f3637-faef-4392-989d-6322d0d42ad4"
} | [
{
"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\nAlfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person t... |
Let $a_1, a_2, \ldots, a_{18}$ be a list of prime numbers such that $\frac{1}{64} \times a_1 \times a_2 \times \cdots \times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18} \frac{1}{\log_{a_i} n}$$ is a positive integer. | 64008420 | 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 $a_1, a_2, \\ldots, a_{18}$ be a list of prime numbers such that $\\frac{1}{64} \\times a_1 \\times a_2 \\times \\... | MATH | {
"ground_truth": "64008420",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "369788d0-1829-4bde-8c71-32b77ceaaaa1"
} | [
{
"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 $a_1, a_2, \\ldots, a_{18}$ be a list of prime numbers such that $\\frac{1}{64} \\times a_1 \\times a_2 \\times \\... |
Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\). | 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 the number of triples \\((a, b, c)\\) of positive integers such that \\(a + ab + abc = 11\\).\n\nRemember to put ... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "78d3bd6d-0057-4c52-8f5d-da7bfa5ce2bc"
} | [
{
"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 triples \\((a, b, c)\\) of positive integers such that \\(a + ab + abc = 11\\).\n\nRemember to put ... |
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