prompt stringlengths 41 2k | solution stringlengths 1 22 | data_source stringclasses 1
value | source_prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | content listlengths 1 1 |
|---|---|---|---|---|---|---|---|
In a triangle $ABC$, points $X$ and $Y$ are on segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY : YC = 2 : 1$. If the area of triangle $AXY$ is $10$, what is the area of triangle $ABC$? | 45 | 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 triangle $ABC$, points $X$ and $Y$ are on segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $... | MATH | {
"ground_truth": "45",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9ef2339c-9123-4980-ae7d-f09962951a59"
} | [
{
"content": "Solve the following math problem step by step. 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 triangle $ABC$, points $X$ and $Y$ are on segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $... |
When simplified, \( (x^{-1} + y^{-1})^{-1} \) is equal to what expression in the form \(\frac{kx}{my}\)? Please find the value of k + m. | 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\nWhen simplified, \\( (x^{-1} + y^{-1})^{-1} \\) is equal to what expression in the form \\(\\frac{kx}{my}\\)? Please f... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2be5d435-479e-4928-ba06-d2e7aea6f70b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhen simplified, \\( (x^{-1} + y^{-1})^{-1} \\) is equal to what expression in the form \\(\\frac{kx}{my}\\)? Please f... |
Circles of radii $5, 5, 8,$ and $\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
| 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\nCircles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime po... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "21f9e06b-89ce-4685-8f2d-491ce440dbe3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCircles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime po... |
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n= \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$? | 502 | 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 $a_1,a_2,\\ldots$ be a sequence determined by the rule $a_n= \\frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_... | MATH | {
"ground_truth": "502",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "872c77ef-1e53-4662-a036-c76cca153b31"
} | [
{
"content": "Solve the following math problem step by step. 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 $a_1,a_2,\\ldots$ be a sequence determined by the rule $a_n= \\frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_... |
Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal.
$$\begin{tabular}{ccccccc}
& P & U & R & P & L & E\\
+ & & C & O & M & E & T \\
\hline
\\
\end{tabular}$$
Find the greatest possible value that the five-digit number $COM... | 98057 | 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 different letter in the following addition represents a different decimal digit. The sum is a six-digit integer w... | MATH | {
"ground_truth": "98057",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6f4497a6-907e-4fd3-9e11-1bcdc56b1dfb"
} | [
{
"content": "Solve the following math problem step by step. 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 different letter in the following addition represents a different decimal digit. The sum is a six-digit integer w... |
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there were three times as many sets of twins as sets of triplets. How many of these $100... | 100 | 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 Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets ac... | MATH | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc24d269-3154-4ef9-941f-9d5abd3a2d37"
} | [
{
"content": "Solve the following math problem step by step. 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 Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets ac... |
Set $B = \{55, 57, 58, 59, 61\}.$ Which number in set B has the smallest prime factor? | 58 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSet $B = \\{55, 57, 58, 59, 61\\}.$ Which number in set B has the smallest prime factor?\n\nRemember to put your answ... | MATH | {
"ground_truth": "58",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "483d6ee1-6eab-4ddb-890a-2f14ab413f6c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSet $B = \\{55, 57, 58, 59, 61\\}.$ Which number in set B has the smallest prime factor?\n\nRemember to put your answ... |
In the figure shown, $AC=13$ and $DC=2$ units. What is the length of the segment $BD$? Express your answer in simplest radical form.
[asy]
draw((0,5)--(12,5));
draw((12,5)--(12,0));
draw((12,0)--(0,5));
draw((12,5)--(10.1,0.8));
draw((10.48,1.64)--(9.47,2.06));
draw((9.47,2.06)--(9.09,1.22));
draw((11,5)--(11,4));
dr... | 23 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the figure shown, $AC=13$ and $DC=2$ units. What is the length of the segment $BD$? Express your answer in simples... | MATH | {
"ground_truth": "23",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "67caae10-fcbb-449c-bec7-fdfa51faacc2"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the figure shown, $AC=13$ and $DC=2$ units. What is the length of the segment $BD$? Express your answer in simples... |
For how many integers $a$ satisfying $1 \le a \le 23$ is it true that $a^{-1} \equiv a \pmod{24}$? | 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\nFor how many integers $a$ satisfying $1 \\le a \\le 23$ is it true that $a^{-1} \\equiv a \\pmod{24}$?\n\nRemember to ... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "52c99808-5b53-4e45-9763-0ec3aa4e7653"
} | [
{
"content": "Solve the following math problem step by step. 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 how many integers $a$ satisfying $1 \\le a \\le 23$ is it true that $a^{-1} \\equiv a \\pmod{24}$?\n\nRemember to ... |
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \(3\) and \(4\) units. In the corner where those sides meet at a right angle, he leaves a small unplanted square \(S\) so that from the air it looks like the right angle symbol. The rest of the field is planted. The s... | 292 | 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\nFarmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \\(3\\) and \\(... | MATH | {
"ground_truth": "292",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5c1a5816-1c9b-49a6-8099-6c1c2322f24a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFarmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths \\(3\\) and \\(... |
Jerry purchased some stock for $ \$14,400$ at the same time that Susan purchased a bond for $ \$6,250$. Jerry’s investment went up $20$ percent the first year, fell $10$ percent the second year, and rose another $20$ percent the third year. Susan’s investment grew at a constant rate of compound interest for three years... | 44 | 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\nJerry purchased some stock for $ \\$14,400$ at the same time that Susan purchased a bond for $ \\$6,250$. Jerry’s inve... | MATH | {
"ground_truth": "44",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4f206627-22a4-44fc-9c82-f0586368c8f8"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nJerry purchased some stock for $ \\$14,400$ at the same time that Susan purchased a bond for $ \\$6,250$. Jerry’s inve... |
Determine the exact value of the series
\[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 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\nDetermine the exact value of the series\n\\[\\frac{1}{5 + 1} + \\frac{2}{5^2 + 1} + \\frac{4}{5^4 + 1} + \\frac{8}{5^8... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "486837d2-df45-40bd-b8ae-da841a9e3f59"
} | [
{
"content": "Solve the following math problem step by step. 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 exact value of the series\n\\[\\frac{1}{5 + 1} + \\frac{2}{5^2 + 1} + \\frac{4}{5^4 + 1} + \\frac{8}{5^8... |
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. Find the area of the rectangle. | 32 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circ... | MATH | {
"ground_truth": "32",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ff55f5f3-3a96-4153-9ee2-9e7a39b155a3"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThree congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circ... |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xy + xz + yz = 1.$ Find the minimum value of $10x^2 + 10y^2 + z^2.$ | 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 $x,$ $y,$ and $z$ be positive real numbers such that $xy + xz + yz = 1.$ Find the minimum value of $10x^2 + 10y^2... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d2ae44d9-2650-4941-a30e-66aaf675b161"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $x,$ $y,$ and $z$ be positive real numbers such that $xy + xz + yz = 1.$ Find the minimum value of $10x^2 + 10y^2... |
Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed:
$\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out.
$\bullet$ No one person can have more shares than the other ... | 315 | 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\nEli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The fol... | MATH | {
"ground_truth": "315",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "441f93a3-8881-4c4d-9239-d567e0ef8b0f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The fol... |
Nathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \times 1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w \times h$ (where $w$ is the number of columns and $h$ is the number of rows), then it costs $w^2$... | 5339 | 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\nNathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \\times 1$... | MATH | {
"ground_truth": "5339",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1f1fa9c8-68c7-4f62-9fee-420defcf1cce"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nNathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \\times 1$... |
Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arran... | 630 | 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\nRobert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one ... | MATH | {
"ground_truth": "630",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cd68b21f-1d2c-424d-b6b3-72ecdf7faf57"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRobert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one ... |
Find the value of \( \frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}} \). The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\). | 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\nFind the value of \\( \\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}} \\). The original answer is in the form... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "789b7499-741c-42c4-9d01-d019ca830d5b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the value of \\( \\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}} \\). The original answer is in the form... |
Triangle ABC has vertices $A(0, 0)$, $B(0, 3)$ and $C(5, 0)$. A point $P$ inside the triangle is $\sqrt{10}$ units from point $A$ and $\sqrt{13}$ units from point $B$. How many units is $P$ from point $C$? Express your answer in simplest radical form.The answer is in the form k\sqrt{m}+n,. Please provide the value of k... | 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\nTriangle ABC has vertices $A(0, 0)$, $B(0, 3)$ and $C(5, 0)$. A point $P$ inside the triangle is $\\sqrt{10}$ units fr... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f6a13bf6-4fc1-4f3c-90ac-c0f6433e0a36"
} | [
{
"content": "Solve the following math problem step by step. 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 vertices $A(0, 0)$, $B(0, 3)$ and $C(5, 0)$. A point $P$ inside the triangle is $\\sqrt{10}$ units fr... |
In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), please find the value of \(k\). | 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\nIn \\(\\triangle ABC\\), we have \\(AB = 1\\) and \\(AC = 2\\). Side \\(\\overline{BC}\\) and the median from \\(A\\) ... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bf53b74b-b5c8-435a-b1b0-2caebb5d0429"
} | [
{
"content": "Solve the following math problem step by step. 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\\), we have \\(AB = 1\\) and \\(AC = 2\\). Side \\(\\overline{BC}\\) and the median from \\(A\\) ... |
Determine the smallest positive integer $n$ with the following property: for every triple of positive integers $x, y, z$, with $x$ dividing $y^3$, $y$ dividing $z^3$, and $z$ dividing $x^3$, it also holds that $(xyz)$ divides $(x + y + z)^n$. | 13 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDetermine the smallest positive integer $n$ with the following property: for every triple of positive integers $x, y, ... | MATH | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e51a6c07-96e6-47ad-8d9d-d27c0db8eccd"
} | [
{
"content": "Solve the following math problem step by step. 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 smallest positive integer $n$ with the following property: for every triple of positive integers $x, y, ... |
Convert the base-$64$ number $100_{64}$ to base $62$. | 144 | 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\nConvert the base-$64$ number $100_{64}$ to base $62$.\n\nRemember to put your answer on its own line after \"Answer:\"... | MATH | {
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "22618727-5c86-46ad-902b-ace054dd2daa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConvert the base-$64$ number $100_{64}$ to base $62$.\n\nRemember to put your answer on its own line after \"Answer:\"... |
Let $f(x) = 1 + x + x^2 + \cdots + x^{100}$. Find $f'(1)$. | 5050 | 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 $f(x) = 1 + x + x^2 + \\cdots + x^{100}$. Find $f'(1)$.\n\nRemember to put your answer on its own line after \"Ans... | MATH | {
"ground_truth": "5050",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5236e690-f702-4dc2-9751-d075a1fc0344"
} | [
{
"content": "Solve the following math problem step by step. 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 $f(x) = 1 + x + x^2 + \\cdots + x^{100}$. Find $f'(1)$.\n\nRemember to put your answer on its own line after \"Ans... |
A pen costs $\text{Rs.}\, 13$ and a notebook costs $\text{Rs.}\, 17$. A school spends exactly $\text{Rs.}\, 10000$ in the year 2017-18 to buy $x$ pens and $y$ notebooks such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in 2018-19, the school spends a little more than $\text{Rs.}\, 10... | 40 | 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 pen costs $\\text{Rs.}\\, 13$ and a notebook costs $\\text{Rs.}\\, 17$. A school spends exactly $\\text{Rs.}\\, 1000... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4f3d5190-02dd-49f1-8712-1319d9a65001"
} | [
{
"content": "Solve the following math problem step by step. 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 pen costs $\\text{Rs.}\\, 13$ and a notebook costs $\\text{Rs.}\\, 17$. A school spends exactly $\\text{Rs.}\\, 1000... |
Rose has a spherical plum of radius 2 and a spherical watermelon of radius 8. She builds a glass sphere around the two fruits to contain them, making the sphere as small as possible. When she has done this, the volume contained within the sphere, but not in the plum or the watermelon, is $K\pi$. What is the value of... | 640 | 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\nRose has a spherical plum of radius 2 and a spherical watermelon of radius 8. She builds a glass sphere around the tw... | MATH | {
"ground_truth": "640",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "536a9d71-e44e-484c-b619-9c2c3254a1a6"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nRose has a spherical plum of radius 2 and a spherical watermelon of radius 8. She builds a glass sphere around the tw... |
How many positive integers are less than $100\pi$? | 314 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers are less than $100\\pi$?\n\nRemember to put your answer on its own line after \"Answer:\"."... | MATH | {
"ground_truth": "314",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "39da4023-e1fd-42ff-8dfb-69107e9208dd"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers are less than $100\\pi$?\n\nRemember to put your answer on its own line after \"Answer:\"."... |
In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17. What is the area of triangle $ABC$?
| 110 | 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 $ABC$, $\\tan \\angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17... | MATH | {
"ground_truth": "110",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "56a368fe-51a8-4879-9b96-053ea9485fea"
} | [
{
"content": "Solve the following math problem step by step. 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 $ABC$, $\\tan \\angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17... |
Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other... | 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\nSeongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shir... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "28116c01-2dfb-4d6b-8ca8-005d349400bf"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSeongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shir... |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? Provide your answer in the form $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b. | 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\nSix regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$? Pr... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ae81147d-7cd2-45f0-a7dc-a9efd6d92c47"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSix regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\\triangle{ABC}$? Pr... |
Find the number of sets $A$ that satisfy the following conditions:
1. $A$ is a set of two positive integers.
2. Each number in $A$ is at least $22\%$ the size of the other number.
3. $A$ contains the number $30$. | 129 | 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 sets $A$ that satisfy the following conditions:\n\n1. $A$ is a set of two positive integers.\n2. Ea... | MATH | {
"ground_truth": "129",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "4284314d-2ed2-436d-99e9-92cb5f57955b"
} | [
{
"content": "Solve the following math problem step by step. 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 sets $A$ that satisfy the following conditions:\n\n1. $A$ is a set of two positive integers.\n2. Ea... |
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Mich... | 144 | 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\nMichel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an... | MATH | {
"ground_truth": "144",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "08971c55-b4d4-45b0-a9b8-6ad51e526d36"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nMichel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an... |
Given that the line $y=kx+b$ intersects the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and that $|AB|=|BC|=2$, find the integer value of $k$. | 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 that the line $y=kx+b$ intersects the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and that $|AB|=|BC|=... | MATH | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "514e669b-d742-439f-bba3-aae7fd7c73dc"
} | [
{
"content": "Solve the following math problem step by step. 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 line $y=kx+b$ intersects the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and that $|AB|=|BC|=... |
Consider the following $100$ sets of $10$ elements each:
\begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*}
Find how many of these sets contain exactly two multiples of $7$. | 42 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nConsider the following $100$ sets of $10$ elements each:\n\\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\... | MATH | {
"ground_truth": "42",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "bfff961a-8bd0-44b4-8e68-23b8ea24480c"
} | [
{
"content": "Solve the following math problem step by step. 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 following $100$ sets of $10$ elements each:\n\\begin{align*} &\\{1,2,3,\\ldots,10\\}, \\\\ &\\{11,12,13,\... |
Equilateral triangle $\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of ... | 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\nEquilateral triangle $\\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\\overline{BC}$ such that $BD = ... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "12aedaf5-36f3-4bba-9cd5-8b430a03e2e9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEquilateral triangle $\\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\\overline{BC}$ such that $BD = ... |
A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees. | 63 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA triangle $\\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point i... | MATH | {
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb7ccb61-0379-4188-b2e1-f0614f0cbfe0"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA triangle $\\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point i... |
Find the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when divided by $9$ is $1$. | 208 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when di... | MATH | {
"ground_truth": "208",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "dd84a1de-27c4-43c4-beba-bf678185b077"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when di... |
Let $f(n) = \sum_{gcd(k,n)=1,1\le k\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_1 p^{e_2}_2 ... p^{e_k}_k$, find $\sum^k_{i=1} p_ie_i$. | 818 | 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 $f(n) = \\sum_{gcd(k,n)=1,1\\le k\\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_... | MATH | {
"ground_truth": "818",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7cfeff76-fb32-4744-8d19-def50927c0ab"
} | [
{
"content": "Solve the following math problem step by step. 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 $f(n) = \\sum_{gcd(k,n)=1,1\\le k\\le n}k^3$ . If the prime factorization of $f(2020)$ can be written as $p^{e_1}_... |
Compute the number of ordered quadruples of positive integers $(a, b, c, d)$ such that
\[ a! \cdot b! \cdot c! \cdot d! = 24! \] | 28 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCompute the number of ordered quadruples of positive integers $(a, b, c, d)$ such that \n\\[ a! \\cdot b! \\cdot c! \\... | MATH | {
"ground_truth": "28",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c426328b-426b-4578-b8a0-7b379c82244d"
} | [
{
"content": "Solve the following math problem step by step. 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 number of ordered quadruples of positive integers $(a, b, c, d)$ such that \n\\[ a! \\cdot b! \\cdot c! \\... |
Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value of $a+b+c+d+e$? | 153 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value o... | MATH | {
"ground_truth": "153",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "49ca9572-cba3-49fb-8164-e1da35671da1"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value o... |
Given $\tan \theta = 5,$ find
\[\frac{1 - \cos \theta}{\sin \theta} - \frac{\sin \theta}{1 + \cos \theta}.\] | 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\nGiven $\\tan \\theta = 5,$ find\n\\[\\frac{1 - \\cos \\theta}{\\sin \\theta} - \\frac{\\sin \\theta}{1 + \\cos \\theta... | MATH | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "854dfbab-767b-45bd-aca7-ea84a85f60a5"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\tan \\theta = 5,$ find\n\\[\\frac{1 - \\cos \\theta}{\\sin \\theta} - \\frac{\\sin \\theta}{1 + \\cos \\theta... |
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [asy]
path box=(0,0)--(1,0)--(1,1.5)--(0,1.5)... | 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\nBarry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the... | MATH | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "63372486-1fd9-4240-a367-309491a5b480"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nBarry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the... |
In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 12$, $\tan Z = 1.5$, and $\tan X = 3$, then what is the area of $WXYZ$?
[asy]
pair WW,X,Y,Z;
Z = (0,0);
Y = (12,0);
WW = (12,18);
X= (18,18);
draw(WW--X--Y--Z--WW);
label("$W$",... | 162 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, $WXYZ$ is a trapezoid such that $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\p... | MATH | {
"ground_truth": "162",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a633ee15-68e7-4f8d-a2e6-1ecd7ef78525"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, $WXYZ$ is a trapezoid such that $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\p... |
To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not flipped or rotated) on his plate, and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed? | 1092 | 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\nTo celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the fou... | MATH | {
"ground_truth": "1092",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5fb80f05-1ccb-4021-9caa-6704618f358d"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTo celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the fou... |
Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^\circ$. Let $M$ be the foot of the altitude from $C$, and let $N$ be a point on the side $AC$ such that $CN = \frac{1}{2} BC$. Determine the measure of the angle $AMN$. | 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\nLet $ABC$ be an isosceles triangle such that $AB = AC$ and $\\angle A = 20^\\circ$. Let $M$ be the foot of the altitud... | MATH | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "84e019b2-a752-4e55-93b4-e6579ddd5a48"
} | [
{
"content": "Solve the following math problem step by step. 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 an isosceles triangle such that $AB = AC$ and $\\angle A = 20^\\circ$. Let $M$ be the foot of the altitud... |
What is the value of $a+b+c+d+e+f$ for the decimal representation of $\frac{4}{37}+\frac{3}{11}+\frac{23}{9}=2.abcdef\ldots$? | 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\nWhat is the value of $a+b+c+d+e+f$ for the decimal representation of $\\frac{4}{37}+\\frac{3}{11}+\\frac{23}{9}=2.abcd... | MATH | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fc4cc1e9-1f80-4a84-9574-cbd98171060b"
} | [
{
"content": "Solve the following math problem step by step. 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 $a+b+c+d+e+f$ for the decimal representation of $\\frac{4}{37}+\\frac{3}{11}+\\frac{23}{9}=2.abcd... |
The numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? | 432 | 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 numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that h... | MATH | {
"ground_truth": "432",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "662abeae-dc56-4c78-b5e0-a1cfe01331ed"
} | [
{
"content": "Solve the following math problem step by step. 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 numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that h... |
For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. Determine the number of three-digit positive integers $m$ for which $m + t(t(m))$ is odd. | 50 | 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 each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the dig... | MATH | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5876ff2f-415c-4b7f-84c6-11cd2c6f2fb3"
} | [
{
"content": "Solve the following math problem step by step. 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 each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the dig... |
There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in th... | 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\nThere are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it take... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "647cb40c-db24-45c2-9752-810674bbed46"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it take... |
For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) + \cos^2(30^\circ +n)$. Evaluate $4\sum^{30}_{n=1} n \cdot T(n).$ | 1395 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor $n$ measured in degrees, let $T(n) = \\cos^2(30^\\circ -n) - \\cos(30^\\circ -n)\\cos(30^\\circ +n) + \\cos^2(30^\... | MATH | {
"ground_truth": "1395",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "51ba29a9-1292-47f1-8c04-daeaf100f818"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFor $n$ measured in degrees, let $T(n) = \\cos^2(30^\\circ -n) - \\cos(30^\\circ -n)\\cos(30^\\circ +n) + \\cos^2(30^\... |
Non-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\angle ABC = \angle BCD = \angle CDA$. Given that $AB = 3$ and $B \neq D$, determine the number of possible lengths for $BC$. | 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\nNon-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\\angle... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fa6b1eb7-b2b2-4eb2-8e37-32f3142b210f"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nNon-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\\angle... |
Find the area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively. | 400 | 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 area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectiv... | MATH | {
"ground_truth": "400",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "241806a7-1da1-4b91-8e13-db41caac9d8f"
} | [
{
"content": "Solve the following math problem step by step. 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 area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectiv... |
Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Comput... | 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\nCircle $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tang... | MATH | {
"ground_truth": "17",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1f745861-980e-4821-bd3e-469424a1405d"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\\omega$ such that $OA=13$. The two tang... |
The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$.... | 3 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "374f27cd-2f00-43cd-ab54-d114f4712ce1"
} | [
{
"content": "Solve the following math problem step by step. 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 circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point... |
Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 698 | 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 cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 t... | MATH | {
"ground_truth": "698",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "cb5eb26a-3d2b-422c-b584-c64537524315"
} | [
{
"content": "Solve the following math problem step by step. 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 cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 t... |
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b - a$? | 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\nThe point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\\circ$ around the point $(1,5)$ and the... | MATH | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "ef03af11-52e0-4193-a3f3-84706b7a299c"
} | [
{
"content": "Solve the following math problem step by step. 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 point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\\circ$ around the point $(1,5)$ and the... |
Let $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of
\[ \sum_{i=1}^{5} |a_i - i| = |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + |a_4 - 4| + |a_5 - 5|. \] | 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\nLet $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected v... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "b9ec9eef-a38a-4a1d-a749-90f9094f359f"
} | [
{
"content": "Solve the following math problem step by step. 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 $(a_1, a_2, a_3, a_4, a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected v... |
In $\Delta ABC$, $AC = BC$, $m\angle DCB = 40^{\circ}$, and $CD \parallel AB$. What is the number of degrees in $m\angle ECD$?
[asy] pair A,B,C,D,E; B = dir(-40); A = dir(-140); D = (.5,0); E = .4 * dir(40);
draw(C--B--A--E,EndArrow); draw(C--D,EndArrow);
label("$A$",A,W); label("$C$",C,NW);label("$B$",B,E);label("$D$... | 40 | 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 $\\Delta ABC$, $AC = BC$, $m\\angle DCB = 40^{\\circ}$, and $CD \\parallel AB$. What is the number of degrees in $m... | MATH | {
"ground_truth": "40",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "a551cc9a-b029-4caf-90e5-d76246927c56"
} | [
{
"content": "Solve the following math problem step by step. 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 $\\Delta ABC$, $AC = BC$, $m\\angle DCB = 40^{\\circ}$, and $CD \\parallel AB$. What is the number of degrees in $m... |
A palindrome is a number which reads the same forward as backward. If a three-digit palindrome is randomly chosen, what is the probability that it is a multiple of 3?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 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 palindrome is a number which reads the same forward as backward. If a three-digit palindrome is randomly chosen, wha... | MATH | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "7f82493a-94cb-4a9e-9e24-20876e9f8bd9"
} | [
{
"content": "Solve the following math problem step by step. 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 palindrome is a number which reads the same forward as backward. If a three-digit palindrome is randomly chosen, wha... |
Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the sho... | 336 | 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\nCarl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, a... | MATH | {
"ground_truth": "336",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d301099b-dfc7-4a20-ba98-10d6280ccc4e"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCarl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, a... |
The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
| 143 | 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 measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer val... | MATH | {
"ground_truth": "143",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "46d5c2d3-b2b1-4a2c-8575-61d9b7b19600"
} | [
{
"content": "Solve the following math problem step by step. 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 measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer val... |
What is the sum of all positive integers that have twice as many digits when written in base $2$ as they have when written in base $3$? Express your answer in base $10$. | 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\nWhat is the sum of all positive integers that have twice as many digits when written in base $2$ as they have when wri... | MATH | {
"ground_truth": "10",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d84887cd-68d9-4112-a60b-0f4c903cd86b"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nWhat is the sum of all positive integers that have twice as many digits when written in base $2$ as they have when wri... |
A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the total bill is $9.45$, find the age of the youngest child. | 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\nA father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for t... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "453ac0c2-291b-4a5c-b7e8-771dbb041f62"
} | [
{
"content": "Solve the following math problem step by step. 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 father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for t... |
Let \(f(x)\) be a real-valued function satisfying \(af(x) + bf(-x) = px^2 + qx + r\). Here, \(a\) and \(b\) are distinct real numbers, and \(p\), \(q\), \(r\) are non-zero real numbers. Determine the condition under which \(f(x) = 0\) will have real solutions. The original answer is in the form \(\left(\frac{k(a+b)}{m(... | 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 real-valued function satisfying \\(af(x) + bf(-x) = px^2 + qx + r\\). Here, \\(a\\) and \\(b\\) ar... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e440553b-1f00-43aa-8431-a7a4cd99b19e"
} | [
{
"content": "Solve the following math problem step by step. 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 real-valued function satisfying \\(af(x) + bf(-x) = px^2 + qx + r\\). Here, \\(a\\) and \\(b\\) ar... |
Positive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \[
2x + y = 2003\quad\text{and}\quad y = |x-a| + |x-b| + |x-c|
\]has exactly one solution. What is the minimum value of $c$? | 1002 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPositive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \\[\n2x + y = 2003\\quad\\... | MATH | {
"ground_truth": "1002",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d2b6b287-d73f-4d51-bb1a-6efd785b9161"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nPositive integers $a$, $b$, and $c$ are chosen so that $a<b<c$, and the system of equations \\[\n2x + y = 2003\\quad\\... |
For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers? | 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\nFor how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers?\n\nRemember to... | MATH | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0c0d0cb9-d069-4b7b-9a4a-4ad69a7c97cf"
} | [
{
"content": "Solve the following math problem step by step. 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 how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers?\n\nRemember to... |
Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$? | 2008 | 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\nCall a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x... | MATH | {
"ground_truth": "2008",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "388e9084-14cf-43b0-b4b0-5b689fa26da9"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nCall a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x... |
Say that a 4-digit positive integer is \textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits. For example, 2013 is mixed. How many 4-digit positive integers are mixed? | 1680 | 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\nSay that a 4-digit positive integer is \\textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the ... | MATH | {
"ground_truth": "1680",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "9dafb525-55ce-4da6-945a-16549eb0aad4"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSay that a 4-digit positive integer is \\textit{mixed} if it has 4 distinct digits, its leftmost digit is neither the ... |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 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\nConsider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin c... | MATH | {
"ground_truth": "118",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "e21d883e-d544-4bbf-ba44-90b7d3945659"
} | [
{
"content": "Solve the following math problem step by step. 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 parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin c... |
The largest term in the binomial expansion of $(1 + \tfrac{1}{2})^{31}$ is of the form $\tfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $b$? | 1024 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe largest term in the binomial expansion of $(1 + \\tfrac{1}{2})^{31}$ is of the form $\\tfrac{a}{b}$, where $a$ and... | MATH | {
"ground_truth": "1024",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "d0355632-2146-4ceb-8bd0-4f806f820d1b"
} | [
{
"content": "Solve the following math problem step by step. 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 largest term in the binomial expansion of $(1 + \\tfrac{1}{2})^{31}$ is of the form $\\tfrac{a}{b}$, where $a$ and... |
Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers? | 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\nTwo integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. W... | MATH | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "440cee3c-cc0c-4a9a-8446-92be67e44e97"
} | [
{
"content": "Solve the following math problem step by step. 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 integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. W... |
Evaluate
$$\begin{aligned}
&\begin{aligned}&~~~~\quad1\left(\frac11+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right)\end{aligned} \\
&+\quad3\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}\right) \\
&+\quad5\left(\f... | 55 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nEvaluate \n \n$$\\begin{aligned}\n&\\begin{aligned}&~~~~\\quad1\\left(\\frac11+\\frac12+\\frac13+\\frac14+\\frac15+\... | MATH | {
"ground_truth": "55",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0fbe38d3-fd40-4ce7-a325-9be8552f4870"
} | [
{
"content": "Solve the following math problem step by step. 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 \n \n$$\\begin{aligned}\n&\\begin{aligned}&~~~~\\quad1\\left(\\frac11+\\frac12+\\frac13+\\frac14+\\frac15+\... |
The sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. Given that $x = 60$ and $y = 63$, determine the length of the largest side of the triangle. | 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\nThe sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \\frac{2\\Delta}{x} = y + \\frac{2\\Delta}{y}$, ... | MATH | {
"ground_truth": "87",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "04fdc1e8-e588-485b-8768-7cc1370055cd"
} | [
{
"content": "Solve the following math problem step by step. 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 sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \\frac{2\\Delta}{x} = y + \\frac{2\\Delta}{y}$, ... |
A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$? | 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\nA positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessar... | MATH | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "837399cf-67d9-4a3b-8b70-3dd641d129a6"
} | [
{
"content": "Solve the following math problem step by step. 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 positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessar... |
If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$. | 4026 | 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_1, a_2, \\ldots, a_{2013} \\in [-2, 2]$ and $a_1 + a_2 + \\cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a... | MATH | {
"ground_truth": "4026",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c9decaad-fc2d-4bb0-affc-b5eb8f0e719e"
} | [
{
"content": "Solve the following math problem step by step. 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_1, a_2, \\ldots, a_{2013} \\in [-2, 2]$ and $a_1 + a_2 + \\cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a... |
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term? | 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\nIn a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth ter... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "112043ac-7e32-412d-a707-bfee048d129a"
} | [
{
"content": "Solve the following math problem step by step. 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 sequence of positive integers, each term after the second is the product of the previous two terms. The sixth ter... |
Suppose that positive integers $m, n, k$ satisfy the equations $$m^2 + 1 = 2n^2,$$ $$2m^2 + 1 = 11k^2.$$ Find the residue when $n$ is divided by $17$. | 5 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that positive integers $m, n, k$ satisfy the equations $$m^2 + 1 = 2n^2,$$ $$2m^2 + 1 = 11k^2.$$ Find the resi... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "c9d16f49-28f2-447a-bf43-1960b82cff2a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nSuppose that positive integers $m, n, k$ satisfy the equations $$m^2 + 1 = 2n^2,$$ $$2m^2 + 1 = 11k^2.$$ Find the resi... |
How many numbers $\overline{abc}$ with $a, b, c > 0$ exist such that $$\overline{cba} \mid \overline{abc}$$?
Note: The vertical line denotes that $\overline{cba}$ divides $\overline{abc}.$ | 81 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many numbers $\\overline{abc}$ with $a, b, c > 0$ exist such that $$\\overline{cba} \\mid \\overline{abc}$$? \n\nN... | MATH | {
"ground_truth": "81",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "fdb764f1-ceba-4606-bad0-832d433474ab"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many numbers $\\overline{abc}$ with $a, b, c > 0$ exist such that $$\\overline{cba} \\mid \\overline{abc}$$? \n\nN... |
A $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes ? | 768 | 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 $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal di... | MATH | {
"ground_truth": "768",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "0b5c15dd-c7d9-483a-9676-7130fab5ad28"
} | [
{
"content": "Solve the following math problem step by step. 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 $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal di... |
There are three men and eleven women taking a dance class. In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two? | 103950 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are three men and eleven women taking a dance class. In how many different ways can each man be paired with a w... | MATH | {
"ground_truth": "103950",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "672a499b-629b-43c9-adff-82ead0190caa"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThere are three men and eleven women taking a dance class. In how many different ways can each man be paired with a w... |
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation
\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \]
are plotted on the complex plane (i.e., using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral... | 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\nLet $r$ be a nonzero real number. The values of $z$ which satisfy the equation\n\\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9... | MATH | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "39dff675-ec1a-41d8-9495-4c30a0a71fab"
} | [
{
"content": "Solve the following math problem step by step. 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 $r$ be a nonzero real number. The values of $z$ which satisfy the equation\n\\[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9... |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}$, $\angle B = 75^{\circ}$, $\angle C = 45^{\circ}$, let $H$ be its orthocenter, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$... | 1132 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven a triangle $ABC$ with angles $\\angle A = 60^{\\circ}$, $\\angle B = 75^{\\circ}$, $\\angle C = 45^{\\circ}$, le... | MATH | {
"ground_truth": "1132",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "caffb7a5-40b1-4953-b52a-b6177c263530"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nGiven a triangle $ABC$ with angles $\\angle A = 60^{\\circ}$, $\\angle B = 75^{\\circ}$, $\\angle C = 45^{\\circ}$, le... |
The sum of a 15-term arithmetic series with first term 7 is $-210$. What is the common difference? | 3 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nThe sum of a 15-term arithmetic series with first term 7 is $-210$. What is the common difference?\n\nRemember to pu... | MATH | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "3965e8e5-dac1-4242-a86a-90bcd0e12902"
} | [
{
"content": "Solve the following math problem step by step. 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 sum of a 15-term arithmetic series with first term 7 is $-210$. What is the common difference?\n\nRemember to pu... |
In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?
[asy]
size(340);
int i, j;
for(i = 0; i<10; i = i+1) {
for(j = 0; j<5; j = j+1) {
if(10*j + i == 11 || 10*j + i == 12 || 10*j + i =... | 372 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to... | MATH | {
"ground_truth": "372",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "268668e3-019f-46d3-9a03-5f46f7e41c8a"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nIn the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to... |
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these? | 33 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combinatio... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "332bcad4-2005-447f-acf2-6fa345204863"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nHow many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combinatio... |
A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^2+y^2<1\) in its interior. Compute the largest possible radius of the circle. | 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\nA circle contains the points \\((0, 11)\\) and \\((0, -11)\\) on its circumference and contains all points \\((x, y)\\... | MATH | {
"ground_truth": "61",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5b00981d-490f-4f70-bbd8-1c4fb0281963"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA circle contains the points \\((0, 11)\\) and \\((0, -11)\\) on its circumference and contains all points \\((x, y)\\... |
For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$.
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number. | 2000000000000000000000 | 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 natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number ... | MATH | {
"ground_truth": "2000000000000000000000",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "378cda86-23af-47f5-b25a-1e45eba910c1"
} | [
{
"content": "Solve the following math problem step by step. 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 natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number ... |
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string? | 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\nStarting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probab... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "f9e13b55-d930-4454-821a-11fa199c753e"
} | [
{
"content": "Solve the following math problem step by step. 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 with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probab... |
Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$. | 85 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\\angle CAB = 80^{\\circ}$ and $\\a... | MATH | {
"ground_truth": "85",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "637417d3-b43f-48bc-a62d-1cb7bc6f5399"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nLet $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\\angle CAB = 80^{\\circ}$ and $\\a... |
If $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, find the area of quadrilateral $BEIH$. The original answer is in \(\frac{k}{m}\) format, ple... | 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\nIf $ABCD$ is a $2 \\times 2$ square, $E$ is the midpoint of $\\overline{AB}$, $F$ is the midpoint of $\\overline{BC}$,... | MATH | {
"ground_truth": "22",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "61984605-cf42-4866-848b-7672b78ceabf"
} | [
{
"content": "Solve the following math problem step by step. 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 $ABCD$ is a $2 \\times 2$ square, $E$ is the midpoint of $\\overline{AB}$, $F$ is the midpoint of $\\overline{BC}$,... |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? | 215 | 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 that is not the sum of a positive integral multiple of $42$ and a positive compos... | MATH | {
"ground_truth": "215",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "07dd2b9a-7cb0-4a98-9878-d7d18159746f"
} | [
{
"content": "Solve the following math problem step by step. 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 that is not the sum of a positive integral multiple of $42$ and a positive compos... |
Define a $\mathrm{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by $A$. How many seve... | 192 | 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\nDefine a $\\mathrm{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of ... | MATH | {
"ground_truth": "192",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "330c9a89-9629-4a80-a45d-09afe128df57"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nDefine a $\\mathrm{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of ... |
A sequence of numbers $a_1,$ $a_2,$ $a_3,$ $a_4,$ $\dots,$ has the property that each term after the first is one less than the product of its two neighbors. If $a_1 = 1492$ and $a_2 = 1776,$ then determine $a_{2003}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3269 | 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 numbers $a_1,$ $a_2,$ $a_3,$ $a_4,$ $\\dots,$ has the property that each term after the first is one les... | MATH | {
"ground_truth": "3269",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "5f833edd-2310-4bad-8b44-f48cc1f59d5a"
} | [
{
"content": "Solve the following math problem step by step. 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 numbers $a_1,$ $a_2,$ $a_3,$ $a_4,$ $\\dots,$ has the property that each term after the first is one les... |
Find the integer part of the number \( \frac{2^1}{1!} + \frac{2^2}{2!} + \cdots + \frac{2^{2018}}{2018!} \). | 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\nFind the integer part of the number \\( \\frac{2^1}{1!} + \\frac{2^2}{2!} + \\cdots + \\frac{2^{2018}}{2018!} \\).\n\n... | MATH | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "70c9ffaa-3842-4177-a213-9814d7566278"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the integer part of the number \\( \\frac{2^1}{1!} + \\frac{2^2}{2!} + \\cdots + \\frac{2^{2018}}{2018!} \\).\n\n... |
A mason has bricks with dimensions $2 \times 5 \times 8$ and other bricks with dimensions $2 \times 3 \times 7$. She also has a box with dimensions $10 \times 11 \times 14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bric... | 24 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nA mason has bricks with dimensions $2 \\times 5 \\times 8$ and other bricks with dimensions $2 \\times 3 \\times 7$. S... | MATH | {
"ground_truth": "24",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "47110c14-6853-4f51-b9ba-69a9968b2ea1"
} | [
{
"content": "Solve the following math problem step by step. 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 mason has bricks with dimensions $2 \\times 5 \\times 8$ and other bricks with dimensions $2 \\times 3 \\times 7$. S... |
Compute the sum of the roots of the equation \((2 - x)^{2005} + x^{2005} = 0\). | 2004 | 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 sum of the roots of the equation \\((2 - x)^{2005} + x^{2005} = 0\\).\n\nRemember to put your answer on it... | MATH | {
"ground_truth": "2004",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "8e352fdf-04c5-4479-8ae3-0e8a7361b749"
} | [
{
"content": "Solve the following math problem step by step. 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 sum of the roots of the equation \\((2 - x)^{2005} + x^{2005} = 0\\).\n\nRemember to put your answer on it... |
If only one quarter of the earth's surface is not covered by bodies of water, and only one half of that exposed land area is inhabitable for humans (because of poles, deserts, etc.), what fraction of the earth's surface can humans live on?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the val... | 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\nIf only one quarter of the earth's surface is not covered by bodies of water, and only one half of that exposed land a... | MATH | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "70cd8310-2cd9-4ad9-8a26-3329038525db"
} | [
{
"content": "Solve the following math problem step by step. 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 only one quarter of the earth's surface is not covered by bodies of water, and only one half of that exposed land a... |
Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(ab + cd = 10\). | 58 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered quadruples of positive integers \\((a,b,c,d)\\) such that \\(ab + cd = 10\\).\n\nRemember t... | MATH | {
"ground_truth": "58",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "34b9918f-a75b-45e2-b3ad-1439c0a88d4c"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nFind the number of ordered quadruples of positive integers \\((a,b,c,d)\\) such that \\(ab + cd = 10\\).\n\nRemember t... |
Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\frac{AW}{BW} = \f... | 33 | math_dapo | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nTwo congruent equilateral triangles $\\triangle ABC$ and $\\triangle DEF$ lie on the same side of line $BC$ so that $B... | MATH | {
"ground_truth": "33",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "6d0b145b-d3bc-4855-b513-58047b718c69"
} | [
{
"content": "Solve the following math problem step by step. 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 congruent equilateral triangles $\\triangle ABC$ and $\\triangle DEF$ lie on the same side of line $BC$ so that $B... |
Ann and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens.
Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann goi... | 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\nAnn and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such... | MATH | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "2bd3f907-4ff0-4285-ab97-041ec8366132"
} | [
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: $Answer (without quotes) where $Answer is the answer to the problem.\n\nAnn and Drew have purchased a mysterious slot machine. Each time it is spun, it chooses a random positive integer such... |
For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$. | 761474 | 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 how many unordered sets $\\{a,b,c,d\\}$ of positive integers, none of which exceed $168$, do there exist integers ... | MATH | {
"ground_truth": "761474",
"style": "rule-lighteval/MATH_v2"
} | {
"index": "1b3309c1-ed72-4632-98a5-f7bf4cf993d3"
} | [
{
"content": "Solve the following math problem step by step. 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 how many unordered sets $\\{a,b,c,d\\}$ of positive integers, none of which exceed $168$, do there exist integers ... |
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