answers listlengths 1 1 | datasets listlengths 1 1 | id stringlengths 64 64 | messages listlengths 1 1 | prompt stringlengths 10 2.99k | reference_answer_raw stringlengths 3 1.22k | source stringclasses 1
value | source_records stringlengths 116 1.08k | verifiability dict |
|---|---|---|---|---|---|---|---|---|
[
"8"
] | [
"deepscaler"
] | a6777e04e8e590dd7d70f9a4e13ce4867c72ceca20eab894ec76166f84fa020a | [
{
"content": "The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.\n$\\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\\\ + & 8 & 2 & 9 & 4 & 3 & 0 \\\\ \\hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \\end{tabular... | The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$ | "8" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:859"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2.2"
] | [
"deepscaler"
] | 485f7a98d8f83e4b148af7e153fa4b3cda69875d4c1fa09434c09b1830df2e6c | [
{
"content": "Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these ... | Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpo... | "2.2" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:861"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{5}"
] | [
"deepscaler"
] | 8c1839df7737bd60756df9578067d92fc2b4a5f01d172348083dd8b68496e151 | [
{
"content": "An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the ... | An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the ur... | "\\frac{1}{5}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:862"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1722"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"-\\frac{1}{40}"
] | [
"deepscaler"
] | 00e2ad057ce509e027eeccc220cc0cc6e4b85896ae5ee6d04d14c4dd41112d82 | [
{
"content": "The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by $\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor$ where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. What is the value of $\... | The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}... | "-\\frac{1}{40}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:863"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1158"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"120"
] | [
"deepscaler"
] | 0aed8d8447590b002bcfe1ac41da16da1d3e8635d3ba5fa33434a4453d94b9e8 | [
{
"content": "Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together?",
"role": "user"
}
] | Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | "120" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:864"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"13 + 4\\sqrt{2}"
] | [
"deepscaler"
] | 2e9ceddb6e48b902b9a6be57ab41e57f7481f4b5c0dd0de03661f7dc67d69f37 | [
{
"content": "The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?",
"role": "user"
}
] | The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$? | "13 + 4\\sqrt{2}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:865"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"16"
] | [
"deepscaler"
] | d75beb3b0ae17df56af4b5e095ffe2d630940d917514f12d289a8e39d452e76f | [
{
"content": "For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$",
"role": "user"
}
] | For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n?$ | "16" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:866"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"125"
] | [
"deepscaler"
] | 6f85417356a87f9ae8a2cd277ea7dff061bb887c87c398d276f983c84e6861b7 | [
{
"content": "Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?",
"role": "user"
}
] | Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? | "125" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:868"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | 21f62728ab848d9b6d3f12b8608d6e33be040a5e68419b68e268a19d515c8a93 | [
{
"content": "Quadrilateral $ABCD$ is inscribed in a circle with $\\angle BAC=70^{\\circ}, \\angle ADB=40^{\\circ}, AD=4,$ and $BC=6$. What is $AC$?",
"role": "user"
}
] | Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is $AC$? | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:869"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"23"
] | [
"deepscaler"
] | 7a0f7efdb74504b621c8d359ff3d0fc039754fd39df4e2356ce7daabf7b05853 | [
{
"content": "A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?",
"role": "user"
}
] | A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | "23" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:870"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1245"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3}{2}\\sqrt3"
] | [
"deepscaler"
] | c8fc9b46f4201323cde02bf3de215a50efb4579ac404c93d061456a4c27a7ae7 | [
{
"content": "Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$?",
"role": "user"
}
] | Let $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2,$ where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$ What is the area of $S$? | "\\frac{3}{2}\\sqrt3" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:871"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"26"
] | [
"deepscaler"
] | eee992cf9cfe2d279c472c590b829745ff8c23e129d7ec2354190230f0da70af | [
{
"content": "For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ ... | For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$? | "26" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:872"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"deepscaler"
] | 14670718c22c9a6d638d0e0f13de37ac9788f53088d1806cb641e9ddc101e808 | [
{
"content": "How many solutions does the equation $\\sin \\left( \\frac{\\pi}2 \\cos x\\right)=\\cos \\left( \\frac{\\pi}2 \\sin x\\right)$ have in the closed interval $[0,\\pi]$?",
"role": "user"
}
] | How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? | "2" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:873"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"597"
] | [
"deepscaler"
] | 105a31ed7cedc2151efdb81c8c7e16242cc465a73d264a0510dff3a8cec3997f | [
{
"content": "How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\\log_b a)^{2017}=\\log_b(a^{2017})?$",
"role": "user"
}
] | How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$ | "597" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:874"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{10-6\\sqrt{2}}{7}"
] | [
"deepscaler"
] | 408c187ba46a7c9055f657c9c03ace660adba1ffce62232e39db9e052847c68f | [
{
"content": "Let $ABC$ be a triangle where $M$ is the midpoint of $\\overline{AC}$, and $\\overline{CN}$ is the angle bisector of $\\angle{ACB}$ with $N$ on $\\overline{AB}$. Let $X$ be the intersection of the median $\\overline{BM}$ and the bisector $\\overline{CN}$. In addition $\\triangle BXN$ is equilatera... | Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$... | "\\frac{10-6\\sqrt{2}}{7}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:875"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"20"
] | [
"deepscaler"
] | 942443b99dd7ce9bb6a4dab5c5d7d5f23f15b78fe01a09262d3253d6942826cc | [
{
"content": "The region consisting of all points in three-dimensional space within 3 units of line segment $\\overline{AB}$ has volume $216\\pi$. What is the length $\\textit{AB}$?",
"role": "user"
}
] | The region consisting of all points in three-dimensional space within 3 units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $\textit{AB}$? | "20" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:876"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"c = a"
] | [
"deepscaler"
] | 9ddda9da827471a0581f959f9412bf6373d0997589132228e4b5d9b6e4c9f04b | [
{
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if:",
"role": "user"
}
] | The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: | "c = a" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:877"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"$k^3 + (k + 1)^3$"
] | [
"deepscaler"
] | 168ad1b3dbd327e6fd3e2c79c3d54b5de996e850d307f7b6370d1a009480ec12 | [
{
"content": "The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as:",
"role": "user"
}
] | The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as: | "$k^3 + (k + 1)^3$" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:878"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"35"
] | [
"deepscaler"
] | b8cfc15ab19ae0f2625a471750737a3c7ebb83a201b9fe739b429fa7df0740a9 | [
{
"content": "The square\n\n$\\begin{tabular}{|c|c|c|} \\hline 50 & \\textit{b} & \\textit{c} \\\\ \\hline \\textit{d} & \\textit{e} & \\textit{f} \\\\ \\hline \\textit{g} & \\textit{h} & 2 \\\\ \\hline \\end{tabular}$\nis a multiplicative magic square. That is, the product of the numbers in each row, column, a... | The square
$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{tabular}$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries ... | "35" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:879"}] | {
"all_reference_components_parse": true,
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"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | 00f0ec83aab457eb8c8b97f7af2f15237a63300e1f7ec46f9a7713d532b1dee3 | [
{
"content": "A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as ma... | A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green... | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:880"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0.21"
] | [
"deepscaler"
] | 64fa432933d207c93b756cebe38b40f1df15433e68f8536fce06a9018a06868c | [
{
"content": "Back in 1930, Tillie had to memorize her multiplication facts from $0 \\times 0$ to $12 \\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the bo... | Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd? | "0.21" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:881"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{26^2}{10}"
] | [
"deepscaler"
] | f86c95811f59c0a3979f51707393edc71a663d1393dac3a850e139b47ff451da | [
{
"content": "Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased?",
"role": "us... | Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? | "\\frac{26^2}{10}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:882"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"167400"
] | [
"deepscaler"
] | af920fa097427ae93395ab7c83e87e5c08b4d4a093912effff5036b0f5a5e4c0 | [
{
"content": "(1901 + 1902 + 1903 + \\cdots + 1993) - (101 + 102 + 103 + \\cdots + 193) =",
"role": "user"
}
] | (1901 + 1902 + 1903 + \cdots + 1993) - (101 + 102 + 103 + \cdots + 193) = | "167400" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:883"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)"
] | [
"deepscaler"
] | d00709e88da40d7628812a94fc55d0db4d62f480f540755ddbd89503b64d4e03 | [
{
"content": "Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ADC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?",
"role": "user"
}
] | Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$? | "\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:884"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2}{5}"
] | [
"deepscaler"
] | 29eb483c60ad9c4aacb0f5b3a7187591af80e1bb3ac3e99e3d3323d525dd1d4f | [
{
"content": "Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?",
"role": "user"
}
] | Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | "\\frac{2}{5}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:885"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1705"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"12"
] | [
"deepscaler"
] | de83ee76a37d11cbe2a95a97dd5f3b52a4cc37eed9022b26fb1472bce30f307f | [
{
"content": "Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three... | Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | "12" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:886"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1009"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | 206603017c46840e907abe3d9780f24a69d301f4e5348453b0424365a44f6049 | [
{
"content": "How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?",
"role": "user"
}
] | How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$? | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:887"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2}{5}"
] | [
"deepscaler"
] | 649a96084c9cadc72ab44f86212e310d011b31a91db751b3df11d71e720fef78 | [
{
"content": "Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the li... | Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup ... | "\\frac{2}{5}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:888"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:2773"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"21"
] | [
"deepscaler"
] | 6ea51dbd572bb28fac8c52b754baccc618632d10da2666ac76a8560a16522cd3 | [
{
"content": "How many $3$-digit positive integers have digits whose product equals $24$?",
"role": "user"
}
] | How many $3$-digit positive integers have digits whose product equals $24$? | "21" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:889"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"900"
] | [
"deepscaler"
] | f2062d4edee35c6a7f0f306555da81d47c7a4d1627b5ddb7e626f088b9288bd9 | [
{
"content": "Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many ... | Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chim... | "900" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:890"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-1"
] | [
"deepscaler"
] | 9239e549c1479c63dd4dc88b039f501f76cf9d772d48e14baf9d90e2930c3c6d | [
{
"content": "For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\\frac{1}{x}-\\frac{1}{y}$ equals",
"role": "user"
}
] | For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals | "-1" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:891"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{625}{144}"
] | [
"deepscaler"
] | 52b09fe2b5d8893858f6348f833bdfdcc31953a086b13362653b59ef3fc89f65 | [
{
"content": "Let $S=\\{(x,y) : x\\in \\{0,1,2,3,4\\}, y\\in \\{0,1,2,3,4,5\\},\\text{ and } (x,y)\\ne (0,0)\\}$. \nLet $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$,... | Let $S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\},\text{ and } (x,y)\ne (0,0)\}$.
Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)=\tan(\angle{CBA})$. Wha... | "\\frac{625}{144}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:892"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac {16!}{2^8}"
] | [
"deepscaler"
] | 93c5681c4386b8cb582c65b36b657387ad5bcf9d1bf48efcc1f6cab5ff0a7f95 | [
{
"content": "A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?",
"role": "user"
}
] | A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? | "\\frac {16!}{2^8}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:893"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{10}{3}"
] | [
"deepscaler"
] | 2ea89988a06710c7d36614a44f90c2e60941cc294ba27e2adaf2e13523bfe440 | [
{
"content": "If $a @ b = \\frac{a \\times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$?",
"role": "user"
}
] | If $a @ b = \frac{a \times b}{a+b}$ for $a,b$ positive integers, then what is $5 @10$? | "\\frac{10}{3}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:894"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"4^9"
] | [
"deepscaler"
] | 02510f0d7086f8c836eeef70a70ac477b433d30fbffc42e79105ff1fbfff3c36 | [
{
"content": "Define $n_a!$ for $n$ and $a$ positive to be\n$n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)$\nwhere $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to",
"role": "user"
}
] | Define $n_a!$ for $n$ and $a$ positive to be
$n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)$
where $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to | "4^9" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:895"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"9"
] | [
"deepscaler"
] | ea75b3053044a3f2866c0466dae34e1391a616ec96eca7bf848e872ed8a86eca | [
{
"content": "Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?",
"role": "user"
}
] | Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school? | "9" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:896"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"25"
] | [
"deepscaler"
] | 1e27f8e8f968ce7e5ca32b0eeabef8310a01cf01c267bad1e112f503d376bcee | [
{
"content": "Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo... | Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $1... | "25" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:897"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"deepscaler"
] | 4a1a8c391b81dbb6b91470b4ef7fb08fe69e3717e07638db195e3208fbea2998 | [
{
"content": "Find the sum of the squares of all real numbers satisfying the equation $x^{256} - 256^{32} = 0$.",
"role": "user"
}
] | Find the sum of the squares of all real numbers satisfying the equation $x^{256} - 256^{32} = 0$. | "8" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:898"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{34}{67}"
] | [
"deepscaler"
] | 88f6c68292b46e8f953213de7c9601a86ba5c1ff07743925915d6f5a5ebb60bd | [
{
"content": "For every $m$ and $k$ integers with $k$ odd, denote by $\\left[ \\frac{m}{k} \\right]$ the integer closest to $\\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that\n\\[\\left[ \\frac{n}{k} \\right] + \\left[ \\frac{100 - n}{k} \\right] = \\left[ \\frac{100}{k} \\right]\\]fo... | For every $m$ and $k$ integers with $k$ odd, denote by $\left[ \frac{m}{k} \right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that
\[\left[ \frac{n}{k} \right] + \left[ \frac{100 - n}{k} \right] = \left[ \frac{100}{k} \right]\]for an integer $n$ randomly chosen from ... | "\\frac{34}{67}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:899"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"deepscaler"
] | b27d8e73e3020361bc7ed284e15cca60ed79c5c30bebfb90ff5b6eebddaa29d2 | [
{
"content": "The number $2013$ is expressed in the form $2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?",
"role": "user"
}
] | The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? | "2" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:900"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1954"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"70"
] | [
"deepscaler"
] | d985ddf8a472c700662527091d003ca836a1cc85d0348cbfca02fa6cb2c28237 | [
{
"content": "The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?",
"role": "user"
}
] | The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? | "70" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:901"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"7"
] | [
"deepscaler"
] | 9df141be63ccd1f0d76dbe2b70fe958875fefae52bbc370ff1c7b0f2984aba03 | [
{
"content": "Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?",
"role": "user"
}
] | Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing? | "7" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:902"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"$30"
] | [
"deepscaler"
] | 21b150d8794e728043e0497d5ab9d3fa526901f75e850841729e9e7515922b2c | [
{
"content": "Alice sells an item at $10 less than the list price and receives $10\\%$ of her selling price as her commission. \nBob sells the same item at $20 less than the list price and receives $20\\%$ of his selling price as his commission. \nIf they both get the same commission, then the list price is",
... | Alice sells an item at $10 less than the list price and receives $10\%$ of her selling price as her commission.
Bob sells the same item at $20 less than the list price and receives $20\%$ of his selling price as his commission.
If they both get the same commission, then the list price is | "$30" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:903"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{3}"
] | [
"deepscaler"
] | 7e906328fdffe758a9832e741e9ab44c8cd99332eb635f4854ba49dc3857c9b0 | [
{
"content": "Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?",
"role": "user"
}
] | Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? | "\\frac{1}{3}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:906"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{\\pi}{3}+1-\\sqrt{3}"
] | [
"deepscaler"
] | 784d16ea5148eaef24046d36671bd5c0a0e3d9c4151b0764b53b347d1a8d9593 | [
{
"content": "A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?",
... | A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$? | "\\frac{\\pi}{3}+1-\\sqrt{3}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:907"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"deepscaler"
] | 503934d1220b454b0d1f52cc14b6486750985aff251ea0b6a13ca8024b690298 | [
{
"content": "Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \\le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$.",
"role": "user"
}
] | Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$. | "3" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:908"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"40"
] | [
"deepscaler"
] | c1f19289bddee7cb27e74501a4ae3de73d60ceef24f03922f0082086b4c1cd55 | [
{
"content": "Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. \nA third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\\triangle APR$ is",
"role": "user"
}
] | Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively.
A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ is | "40" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:909"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"deepscaler"
] | 2554fd9b9bb4ae4820463be354be0a3a560f50d735b9c168eae4ad1c2ad0bac7 | [
{
"content": "What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$, each with leading coefficient 1?",
"role": "user"
}
] | What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$, each with leading coefficient 1? | "3" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:910"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"10"
] | [
"deepscaler"
] | 912f272e4749e1af6206167ebbaaf56d9e771de5833efdef9aabb6e196a6a613 | [
{
"content": "A sign at the fish market says, \"50% off, today only: half-pound packages for just $3 per package.\" What is the regular price for a full pound of fish, in dollars?",
"role": "user"
}
] | A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? | "10" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:911"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{243}{1024}"
] | [
"deepscaler"
] | a62da91dc1cd904e10952a5901dec3cd573f1f4bc874c9f8831475897e4ecfe7 | [
{
"content": "An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?",
"role": "user"
}
] | An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black? | "\\frac{243}{1024}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:912"}] | {
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} |
[
"1"
] | [
"deepscaler"
] | a086763a799b7de5081f48360cde64f5f7c40d12cfaf2cc55a4710cdc7d6df22 | [
{
"content": "Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person pic... | Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)
The number pick... | "1" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:913"}] | {
"all_reference_components_parse": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"80"
] | [
"deepscaler"
] | 7ad75f12ac2fb80f2c904183041c1304604d3a2cb319e074e3540a84b754e8ad | [
{
"content": "Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\\{0,1,2\\}$. How many equilateral triangles have all their vertices in $S$?",
"role": "user"
}
] | Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles have all their vertices in $S$? | "80" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:914"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{20}"
] | [
"deepscaler"
] | 2c0d8834b6e132138449edf16949d8013ec42c892a756d3ce45928653513bb76 | [
{
"content": "A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?",
"role": "user"
}
] | A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? | "\\frac{1}{20}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:915"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"deepscaler"
] | c22d965a3e94e9a89a8f60b6e89c2dd18b6dd5ac6c57512db4783b44c28d73df | [
{
"content": "Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\\frac{700}{99}= 7.\\overline{07}= 7.070707\\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special num... | Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\frac{700}{99}= 7.\overline{07}= 7.070707\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special numbers? | "8" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:916"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | f7828ced32d67459c22b1e4bbab511e35e45a832b177e0cfa2cdd7cd4d765756 | [
{
"content": "For how many three-digit whole numbers does the sum of the digits equal $25$?",
"role": "user"
}
] | For how many three-digit whole numbers does the sum of the digits equal $25$? | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:917"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"$y=100-5x-5x^{2}$"
] | [
"deepscaler"
] | 6a1da732b53e9d48cd217f6684acd4e90c344e03b79cbcf8af20c72fefd01d15 | [
{
"content": "The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:\n\\[\\begin{tabular}[t]{|c|c|c|c|c|c|}\\hline x&0&1&2&3&4\\\\\\hline y&100&90&70&40&0\\\\\\hline\\end{tabular}\\]",
"role": "user"
}
] | The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[\begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline\end{tabular}\] | "$y=100-5x-5x^{2}$" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:918"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{2}"
] | [
"deepscaler"
] | 439eaab2ee29dad2a2b604e261259093fdf9d54600a6813f850fb189d9a913bd | [
{
"content": "Points $A$ and $C$ lie on a circle centered at $O$, each of $\\overline{BA}$ and $\\overline{BC}$ are tangent to the circle, and $\\triangle ABC$ is equilateral. The circle intersects $\\overline{BO}$ at $D$. What is $\\frac{BD}{BO}$?",
"role": "user"
}
] | Points $A$ and $C$ lie on a circle centered at $O$, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$. What is $\frac{BD}{BO}$? | "\\frac{1}{2}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:919"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2.1"
] | [
"deepscaler"
] | 6b4582feca4aff083c58dfd0adb7287871dcca405477087a7f29f2123aef93e6 | [
{
"content": "On a $4 \\times 4 \\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to",
"role": "user"
}
] | On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to | "2.1" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:920"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"deepscaler"
] | 285a7788d57354a49e82ff1e3be9388ca94ca5d60af3b621747cedea7ce59d36 | [
{
"content": "The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is:",
"role": "user"
}
] | The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is: | "2" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:921"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"36"
] | [
"deepscaler"
] | 796f79f36113fc1ae9c6af9775b58c756cf43e14136134c291cf74a01699782e | [
{
"content": "The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\\sqrt{b}+\\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?",
"role": "user"
}
] | The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$? | "36" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:922"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"45"
] | [
"deepscaler"
] | 91c41d8b14247f97ac6c2e92cc48b5ed427b4c413300ea994e995b24161ef761 | [
{
"content": "Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have?",
"rol... | Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have? | "45" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:923"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"33"
] | [
"deepscaler"
] | 5bb46778e840be1710443fbac798d018c4f57b1e5f018406d7e446801bfbc4ff | [
{
"content": "A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of ... | A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by memb... | "33" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:924"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"5"
] | [
"deepscaler"
] | 62b4fab332602a339c01f104b2aa998f4aa0995790556477e34fb56b6466dcd4 | [
{
"content": "In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the m... | In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of... | "5" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:925"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"200"
] | [
"deepscaler"
] | 50597f26335621adf14bbeea5cecb8dfefbd569be8e129cd7ed88b218bcc457d | [
{
"content": "$2.46 \\times 8.163 \\times (5.17 + 4.829)$ is closest to",
"role": "user"
}
] | $2.46 \times 8.163 \times (5.17 + 4.829)$ is closest to | "200" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:926"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"$25\\%$"
] | [
"deepscaler"
] | 324385a51d38144b1bdc22da80e4bfa8efe459f7bcfc509e02086d4e5e22533c | [
{
"content": "At Typico High School, $60\\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\\%$ say that they dislike it, and the rest say that they like it. What fraction o... | At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | "$25\\%$" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:927"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:2685"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"21"
] | [
"deepscaler"
] | 6db83c4a442196e53aff43412f7a3efba78eacdaa7db6f67a5ea553f46468d41 | [
{
"content": "Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\\circ}$. Thus the three polygons form a new polygon with $A$ as an interior poi... | Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest... | "21" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:928"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{9}{17}"
] | [
"deepscaler"
] | fd94b41c4b25809af13ccbc2a5fa95077c1fbec0b66ca91b5ff90f70fa9fe512 | [
{
"content": "Jefferson Middle School has the same number of boys and girls. $\\frac{3}{4}$ of the girls and $\\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls?",
"role": "user"
}
] | Jefferson Middle School has the same number of boys and girls. $\frac{3}{4}$ of the girls and $\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls? | "\\frac{9}{17}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:929"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"18"
] | [
"deepscaler"
] | b80ee74e62c031c88e930903c81474e040b31c775bed15411853f934135841b4 | [
{
"content": "The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?",
"role": "user"
}
] | The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? | "18" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:930"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"$b > 2$"
] | [
"deepscaler"
] | b7d44e56c2b29a5a6e1be15c924d4ba5824897b82c9c225860a7d0e45279929b | [
{
"content": "The number $121_b$, written in the integral base $b$, is the square of an integer, for",
"role": "user"
}
] | The number $121_b$, written in the integral base $b$, is the square of an integer, for | "$b > 2$" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:931"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"8072"
] | [
"deepscaler"
] | c0e97766aef6a1187585e313712427aa1837c7ba1bc67e35b0f47c2a2d12b88e | [
{
"content": "Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have?",
"role": ... | Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have? | "8072" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:932"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"146"
] | [
"deepscaler"
] | 7ef9bb38d2534bdc413660bd947db5292783362a78a40c9f94a23a08b3bbc39f | [
{
"content": "Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchi... | Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren? | "146" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:933"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
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} |
[
"1462"
] | [
"deepscaler"
] | 073748244c9245ef304f8c2e2a791deec2c8ce4d3e667390fc7eee9f50412e99 | [
{
"content": "A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?",
"role": "user"
}
] | A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | "1462" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:934"}] | {
"all_reference_components_parse": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"10"
] | [
"deepscaler"
] | e9fa57db228bbcf92ac622a44a2c6b860734dfbe69cca1c864514d55b869e618 | [
{
"content": "For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \\ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \\le 50$ is $f_{50}(n) = 12?$",
"role": "user"
}
] | For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | "10" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:935"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:1012"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\pi-3"
] | [
"deepscaler"
] | d4093a654b927d45acccf9839233721d51d6ee0b4535bb659eea2db952330f01 | [
{
"content": "$|3-\\pi|=$",
"role": "user"
}
] | $|3-\pi|=$ | "\\pi-3" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:936"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | 81e820d0e6e83ab85dbfde1e463fc5995d08f15e3108f1d340ce014f70cfe90f | [
{
"content": "What is the value of $\\sqrt{(3-2\\sqrt{3})^2}+\\sqrt{(3+2\\sqrt{3})^2}$?",
"role": "user"
}
] | What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:937"}] | {
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"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2.44"
] | [
"deepscaler"
] | 80f0e847b45a9c075ec2380eb0d35340950bd2adc122af1b640e4ecd5e0e71b7 | [
{
"content": "In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\\pm 0.00312$. \nThe experimenter wishes to announce a value for $C$ in which every digit is significant. \nThat is, whatever $C$ is, the announced value must be the correct result when $C$ is round... | In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$.
The experimenter wishes to announce a value for $C$ in which every digit is significant.
That is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digi... | "2.44" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:938"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"48"
] | [
"deepscaler"
] | d30d8d4afca5179d0e225b0785031e227fd2583d6ee7847329529d9bcb8a8e13 | [
{
"content": "Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:",
"role": "user"
}
] | Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: | "48" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:939"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"10"
] | [
"deepscaler"
] | 008ba2a81887a8099d27ab1e58aaa086b17b2913c659ca9119aebc0c25724e76 | [
{
"content": "The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. ... | The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of males in... | "10" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:940"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"38"
] | [
"deepscaler"
] | 4f296a4bf227fbfc4bb5e3cfe5a88eaa52cfc8daefd7ed8f01ba57a441efa8cc | [
{
"content": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is",
"role": "user"
}
] | The shaded region formed by the two intersecting perpendicular rectangles, in square units, is | "38" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:941"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"64 \\pi"
] | [
"deepscaler"
] | b8d794d4bddda3c7f986a858a339313585f128c8a5e2bc9b887b629a595aa1b8 | [
{
"content": "The two circles pictured have the same center $C$. Chord $\\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\\overline{AD}$ has length $16$. What is the area between the two circles?",
"role": "user"
}
] | The two circles pictured have the same center $C$. Chord $\overline{AD}$ is tangent to the inner circle at $B$, $AC$ is $10$, and chord $\overline{AD}$ has length $16$. What is the area between the two circles? | "64 \\pi" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:942"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{11}{7}"
] | [
"deepscaler"
] | 7caed75c702df0b7da6a6703455cf45f2544e4da556c29b265646995706eda3b | [
{
"content": "What is the value of\n$\\displaystyle \\left(\\left((2+1)^{-1}+1\\right)^{-1}+1\\right)^{-1}+1$?",
"role": "user"
}
] | What is the value of
$\displaystyle \left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1$? | "\\frac{11}{7}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:943"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2}{5}"
] | [
"deepscaler"
] | bbfc9f2602038ccd79bdfd4f7ea0fc222fb87e588d039cddb3d9f13c7cd1c64d | [
{
"content": "A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?",
"role": "user"
}
] | A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? | "\\frac{2}{5}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:944"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"65"
] | [
"deepscaler"
] | f3373d0790f0146c702f35b176ae14dc776ed2dfadfa826223098026d8414b86 | [
{
"content": "Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$,... | Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km... | "65" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:945"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"5"
] | [
"deepscaler"
] | 3a2187c8d17905320c22aa4899cbe377c6dfd481589b765a2424ce444aeaa6a9 | [
{
"content": "For what value of $k$ does the equation $\\frac{x-1}{x-2} = \\frac{x-k}{x-6}$ have no solution for $x$?",
"role": "user"
}
] | For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | "5" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:946"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"147"
] | [
"deepscaler"
] | c8ca6467057a42d3546e4843912d6364d3324e21296275917963d650f10a158f | [
{
"content": "Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. ... | Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon... | "147" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:948"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"29"
] | [
"deepscaler"
] | 06dd5a11b192ae7edadbdffd0735896030143752234b54f572d7b94fbd57d530 | [
{
"content": "For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive integer value of $n$?... | For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$? | "29" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:949"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{12}"
] | [
"deepscaler"
] | ef578d8ec2aa78bb65687a243135a687c639c4ecbf9cd4bbb16b120e02c488e5 | [
{
"content": "What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$",
"role": "user"
}
] | What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ | "\\frac{1}{12}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:950"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3}{4}"
] | [
"deepscaler"
] | 1fdfa71187a6546be3842afeb1b4c5ddcd89b65dfec0bfe11b8a9d828646f5c8 | [
{
"content": "The value of $\\frac {1}{2 - \\frac {1}{2 - \\frac {1}{2 - \\frac12}}}$ is",
"role": "user"
}
] | The value of $\frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}}$ is | "\\frac{3}{4}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:951"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{56}{225}"
] | [
"deepscaler"
] | 786d23ec487e07418b3ab09f0289bfc90df9d558383374e8459ba3e6dd1790ec | [
{
"content": "An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?",
"role": "user"
}
] | An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? | "\\frac{56}{225}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:952"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"deepscaler"
] | 57b03a71301bd60c100118625feb28368254080246c61ab6beab3fa47410d7e3 | [
{
"content": "When the number $2^{1000}$ is divided by $13$, the remainder in the division is",
"role": "user"
}
] | When the number $2^{1000}$ is divided by $13$, the remainder in the division is | "3" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:954"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"$\\frac{15}{8}$"
] | [
"deepscaler"
] | 2eb6f3483634c9a9b168dcd7323c5dd911aa3b7c4a94c3513db19deb39ddada4 | [
{
"content": "A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?",
"role": "user"
}
] | A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | "$\\frac{15}{8}$" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:955"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:2673"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"(-1,2)"
] | [
"deepscaler"
] | 54598c80a74b49bf2cfcfd5adc577fdf35f89849aa161c3fc00ac34558d8fb19 | [
{
"content": "All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?",
"role": "user"
}
] | All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point? | "(-1,2)" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:956"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac {r + t}{r - t}"
] | [
"deepscaler"
] | 345f5ddcf9cd8e18eed8d7409ec81fc39b38583aa48e464e42ec4ed209fb0b44 | [
{
"content": "Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is:",
"role": ... | Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: | "\\frac {r + t}{r - t}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:957"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"c+a"
] | [
"deepscaler"
] | 85c552dd4b0834925d7c4893384931868d1c4b2e4049280eff2307d04b73ee9d | [
{
"content": "The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is:",
"role": "user"
}
] | The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: | "c+a" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:958"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"deepscaler"
] | ea1b96481d0ee86f4da612aa794452429417ac4649ec5f04d3b1fe8f1916eb8c | [
{
"content": "The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?",
"role": "user"
}
] | The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family? | "6" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:959"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"4"
] | [
"deepscaler"
] | 9436d4db3a81b111f3f044562fb75ce34421c4fe95e841db5777d149a1fa4d73 | [
{
"content": "When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list?",
"role": "user"
}
] | When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list? | "4" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:960"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{5}"
] | [
"deepscaler"
] | 48704e009bef4665ff1880bba321cf409716a8fc038bccca2d02c148a0ab6764 | [
{
"content": "A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?",
"role": ... | A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color? | "\\frac{1}{5}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:961"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2(n-1)"
] | [
"deepscaler"
] | cf728a2129d54e8a96e8b0b78d75bb196c7838055a7fe5b74650878bcbae47ca | [
{
"content": "A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:",
... | A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: | "2(n-1)" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:962"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{4}"
] | [
"deepscaler"
] | 0bcf3912f8f7d07b977b5d3449d25d9cdebee5d88c22eb9f1956f1e8b356407d | [
{
"content": "Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability ... | Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell ha... | "\\frac{1}{4}" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:963"}, {"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:2605"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"0"
] | [
"deepscaler"
] | 079ac7f953c40d4148970d0b5fbfbd7c554632f62c88b1c54e52365cf7f4d47d | [
{
"content": "Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$.",
"role": "user"
}
] | Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$. | "0" | deepscaler | [{"dataset": "deepscaler", "extra_info": null, "file": "deepscaler.json", "row_id": "deepscaler:deepscaler.json:964"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
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