answers listlengths 1 1 | datasets listlengths 1 1 | id stringlengths 64 64 | messages listlengths 1 1 | prompt stringlengths 10 3.97k | reference_answer_raw stringlengths 9 3.95k | source stringclasses 11
values | source_records stringlengths 283 1.44k | verifiability dict |
|---|---|---|---|---|---|---|---|---|
[
"45"
] | [
"skywork"
] | ad850944f2060a323bb55eb8c834cf9d6eaf4cdaa35634f71fe893dffeb83bc8 | [
{
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ are perpendicular. An altitude $\\overline{ER}$ from $E$ to line $DF$ is such that $ER = 15$. Given $DP= 27$ and $EQ = 36$, determine the length of ${DF}$.",
"role": "user"
}
] | Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. An altitude $\overline{ER}$ from $E$ to line $DF$ is such that $ER = 15$. Given $DP= 27$ and $EQ = 36$, determine the length of ${DF}$. | "[\"45\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1078, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15091"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"49"
] | [
"skywork"
] | a29a9a5c7b12bb07049c1399d77b2d1f42c93ea88e68ba21079e741889cd0145 | [
{
"content": "Let $\\mathbf{D}$ be the $2 \\times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\\det \\mathbf{D}.$",
"role": "user"
}
] | Let $\mathbf{D}$ be the $2 \times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\det \mathbf{D}.$ | "[\"49\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1081, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15092"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3}{5}"
] | [
"skywork"
] | eee56ed13cfc67ddebc54bf1f0b6ccb658eca6cfda9dac50a6313a8c3484da4c | [
{
"content": "(USAMO 1998) Let $\\Gamma_{1}$ and $\\Gamma_{2}$ be two concentric circles, with $\\Gamma_{2}$ inside $\\Gamma_{1}$. Let $A, B$ be points on $\\Gamma_{1}$ and $\\Gamma_{2}$ respectively such that (AB) is tangent to $\\Gamma_{2}$.\n\nLet $C$ be the second intersection point of $(A B)$ with $\\Gamma... | (USAMO 1998) Let $\Gamma_{1}$ and $\Gamma_{2}$ be two concentric circles, with $\Gamma_{2}$ inside $\Gamma_{1}$. Let $A, B$ be points on $\Gamma_{1}$ and $\Gamma_{2}$ respectively such that (AB) is tangent to $\Gamma_{2}$.
Let $C$ be the second intersection point of $(A B)$ with $\Gamma_{1}$ and let $D$ be the midpoin... | "[\"\\\\frac{3}{5}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1060, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15093"}] | {
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} |
[
"16"
] | [
"skywork"
] | 231291459fae20027af68d0bd3b1a1c29e948f4b6601e4d05bdcff7fc440eea7 | [
{
"content": "The average of Amy's, Ben's, and Chris's ages is 10. Five years ago, Chris was twice the age of Amy at that time. In 5 years, Ben's age will be half of Amy's age at that time. How old is Chris now?",
"role": "user"
}
] | The average of Amy's, Ben's, and Chris's ages is 10. Five years ago, Chris was twice the age of Amy at that time. In 5 years, Ben's age will be half of Amy's age at that time. How old is Chris now? | "[\"16\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1057, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15094"}] | {
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} |
[
"396"
] | [
"skywork"
] | c84a4fa38de5ccae4058ae6a1a533f3a184e8c4a9f9ff2eab241a6027cdd7fa1 | [
{
"content": "Let $\\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\\phi(n)\\mid n$.",
"role": "user"
}
] | Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$. | "[\"396\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1082, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15095"}] | {
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"label_correctness_independently_verified": false,
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"reference_components": 1,
"reference_present": true
} |
[
"1-2^{-\\frac{1}{n}}"
] | [
"skywork"
] | 746facf5536c9a05caeb774844c64bc9faeaee5974ad3bc64d1c50f152687600 | [
{
"content": "3. For $n$ positive numbers $x_{1}, x_{2}, \\cdots, x_{n}$ whose sum equals 1, let $S$ be the largest of the following numbers: $\\frac{x_{1}}{1+x_{1}}, \\frac{x_{2}}{1+x_{1}+x_{2}}$, $\\cdots, \\frac{x_{n}}{1+x_{1}+x_{2}+\\cdots+x_{n}}$. Find the minimum possible value of $S$, and for what values... | 3. For $n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum equals 1, let $S$ be the largest of the following numbers: $\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}$, $\cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}}$. Find the minimum possible value of $S$, and for what values of $x_{1}, x_{2}, \cdots, x... | "[\"1-2^{-\\\\frac{1}{n}}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1077, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15096"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{6}"
] | [
"skywork"
] | fe87d9d3f342bd864f3a78b79f85c61850aca94656aee05886da9a2f98a73e0d | [
{
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses?",
"role": "user"
}
] | Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? | "[\"\\\\frac{1}{6}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1090, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15097"}] | {
"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
} |
[
"E"
] | [
"skywork"
] | ec8e5668d898eed7733e6b170cac3ac1704e6bee4ef90d2078d8ee501809562e | [
{
"content": "2.25 To satisfy the equation $\\frac{a+b}{a}=\\frac{b}{a+b}$, then $a$ and $b$ must\n(A) both be rational numbers.\n(B) both be real numbers but not rational numbers.\n(C) not both be real numbers.\n(D) one be real, the other not real.\n(E) one real, one not real, or both not real.\n(11th American... | 2.25 To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, then $a$ and $b$ must
(A) both be rational numbers.
(B) both be real numbers but not rational numbers.
(C) not both be real numbers.
(D) one be real, the other not real.
(E) one real, one not real, or both not real.
(11th American High School Mathematics Exami... | "[\"E\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1076, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15098"}] | {
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"label_correctness_independently_verified": false,
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"reference_components": 1,
"reference_present": true
} |
[
"0"
] | [
"skywork"
] | 5e883970ff8347cd934e2f3f734b00b55dc5d35c19cd1cccad126e2cdc17bef4 | [
{
"content": "6 If the inequality $a \\sin ^{2} x+\\cos x \\geqslant a^{2}-1$ holds for any $x \\in \\mathbf{R}$, then the range of real number $a$ is $\\qquad$.",
"role": "user"
}
] | 6 If the inequality $a \sin ^{2} x+\cos x \geqslant a^{2}-1$ holds for any $x \in \mathbf{R}$, then the range of real number $a$ is $\qquad$. | "[\"0\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1088, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15099"}] | {
"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
} |
[
"71"
] | [
"skywork"
] | 1a2d4b542bc51a0a0f957a506375a065923a08923c5eea44e33c7c067e312eb3 | [
{
"content": "Polish A.\n\nChichikov is playing with Nozdryov. First, Nozdryov distributes 1001 nuts among three boxes. After looking at the distribution, Chichikov names any integer \\( N \\) from 1 to 1001. Then Nozdryov must, if necessary, move one or several nuts to an empty fourth box and show Chichikov on... | Polish A.
Chichikov is playing with Nozdryov. First, Nozdryov distributes 1001 nuts among three boxes. After looking at the distribution, Chichikov names any integer \( N \) from 1 to 1001. Then Nozdryov must, if necessary, move one or several nuts to an empty fourth box and show Chichikov one or several boxes where t... | "[\"71\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1072, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15100"}, {"dataset": "skywork", "extra_... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"19"
] | [
"skywork"
] | 5174b9a7ad2c6adeed5ccfdd9682ca9ce624554dc5039ff0f1c9dd3fdd3b0c6e | [
{
"content": "81. Select several numbers from $1,2,3,4,5,6$ so that the sum of the selected numbers is a multiple of 3 but not a multiple of 5. There are $\\qquad$ different ways to do this.",
"role": "user"
}
] | 81. Select several numbers from $1,2,3,4,5,6$ so that the sum of the selected numbers is a multiple of 3 but not a multiple of 5. There are $\qquad$ different ways to do this. | "[\"19\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1079, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 7, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15101"}] | {
"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
} |
[
"x = 0"
] | [
"skywork"
] | 46136ce7b4b7332aaadadd3e6766f7e3558ce0b63937f694fb21cefbdd096171 | [
{
"content": "The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \\quad n\\geq 2\\] For every natural number $n\\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.",
"role": "user"
}
] | The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$. | "[\" x = 0 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1058, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15102"}] | {
"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
} |
[
"4046"
] | [
"skywork"
] | f688def2b860939ca8e24c5aa35d99dc551d34f76102d1e117a6c97d9fbcbe5b | [
{
"content": "Consider a polynomial $P(x) \\in \\mathbb{R}[x]$ , with degree $2023$ , such that $P(\\sin^2(x))+P(\\cos^2(x)) =1$ for all $x \\in \\mathbb{R}$ . If the sum of all roots of $P$ is equal to $\\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ?",
"role": "user"
}
] | Consider a polynomial $P(x) \in \mathbb{R}[x]$ , with degree $2023$ , such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$ . If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$ ? | "[\"4046\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1083, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15105"}, {"dataset": "skywork", "extra_inf... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2^32"
] | [
"skywork"
] | 64a70ce53111814fba5f5badebcb698da6bfa0cc932a9474cab4d68485896223 | [
{
"content": "It is known that a regular \\( n \\)-gon can be constructed with straightedge and compass if \\( n \\) is a prime that is 1 plus a power of 2. It is also possible to construct a \\( 2n \\)-gon whenever an \\( n \\)-gon is constructible, or a \\( p_{1}p_{2} \\cdots p_{m} \\)-gon where the \\( p_{i}... | It is known that a regular \( n \)-gon can be constructed with straightedge and compass if \( n \) is a prime that is 1 plus a power of 2. It is also possible to construct a \( 2n \)-gon whenever an \( n \)-gon is constructible, or a \( p_{1}p_{2} \cdots p_{m} \)-gon where the \( p_{i} \)'s are distinct primes of the a... | "[\"2^32\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1089, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15106"}] | {
"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
} |
[
"24"
] | [
"skywork"
] | 9dd56c1da736f957b2045f5ab5bde6159c1ee3fa9c44b1035f53788765cbf7c4 | [
{
"content": "In each cell of a chessboard, there are two cockroaches. At a certain moment, each cockroach crawls to an adjacent (by side) cell, and the cockroaches that were in the same cell crawl to different cells. What is the maximum number of cells on the board that can remain free after this?",
"role"... | In each cell of a chessboard, there are two cockroaches. At a certain moment, each cockroach crawls to an adjacent (by side) cell, and the cockroaches that were in the same cell crawl to different cells. What is the maximum number of cells on the board that can remain free after this? | "[\"24\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1094, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15107"}, {"dataset": "skywork", "extra_... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2011"
] | [
"skywork"
] | 38fa2371c74c192118f8913b4d9a21d18c48cf1857341e9accb9f4cf7b7c3283 | [
{
"content": "9. Rectangle $A B C D$ has an area of 2011 square centimeters. The vertices of trapezoid $A F G E$ are such that $F$ is on $B C$, and $D$ is the midpoint of the leg $E G$. Find the area of trapezoid $A F G E$.",
"role": "user"
}
] | 9. Rectangle $A B C D$ has an area of 2011 square centimeters. The vertices of trapezoid $A F G E$ are such that $F$ is on $B C$, and $D$ is the midpoint of the leg $E G$. Find the area of trapezoid $A F G E$. | "[\"2011\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1098, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15109"}] | {
"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}{7},0)"
] | [
"skywork"
] | 5c89b87ef2acc46a785c90a5d8b2acde54541a647aa6619eb687b0a48371a1d9 | [
{
"content": "15. Let the ellipse $C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ have its left and right foci as $F_{1}$ and $F_{2}$, respectively, and its right vertex as $A$. Let $P$ be any point on the ellipse $C$. It is known that the maximum value of $\\overrightarrow{P F_{1}} \\cdot \\overrightar... | 15. Let the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ have its left and right foci as $F_{1}$ and $F_{2}$, respectively, and its right vertex as $A$. Let $P$ be any point on the ellipse $C$. It is known that the maximum value of $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}$ is 3, and the... | "[\"(\\\\frac{2}{7},0)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1085, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15110"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"-49"
] | [
"skywork"
] | a20298d517d253ffbec0df3191b77de2771077f5887b2144fc3c65ebaa95878c | [
{
"content": "The function $f$ satisfies \\[\nf(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1\n\\]for all real numbers $x,y$. Determine the value of $f(10)$.",
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}
] | The function $f$ satisfies \[
f(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1
\]for all real numbers $x,y$. Determine the value of $f(10)$. | "[\"-49\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1091, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15111"}] | {
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[
"5"
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"skywork"
] | 45027f9f81bba48e84a5f20efd4273b694bafb5c47e9f1d491c461fb868f7979 | [
{
"content": "A5 The simplified expression $(\\sin x+2 \\cos x)^{2}+(2 \\sin x-\\cos x)^{2}$ is equal to:\n(A) 0\n(B) $3 \\sin x+\\cos x$\n(C) $\\cos 2 x$\n(D) 1\n(E) 5",
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}
] | A5 The simplified expression $(\sin x+2 \cos x)^{2}+(2 \sin x-\cos x)^{2}$ is equal to:
(A) 0
(B) $3 \sin x+\cos x$
(C) $\cos 2 x$
(D) 1
(E) 5 | "[\"5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1105, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15112"}] | {
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"skywork"
] | 895fb4d4aa69b46a68b52e3f10c69003974e3bca6ee95e6009c416e817892677 | [
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"content": "There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals?",
"role": "user"
}
] | There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals? | "[\"696\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1108, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15113"}] | {
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"skywork"
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"content": "If the set $\\{1, a, \\frac{b}{a}\\} = \\{0, a^2, a+b\\}$, find the value of $a^{2015} + b^{2016}$.",
"role": "user"
}
] | If the set $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2016}$. | "[\"-1\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1095, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15114"}] | {
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"-63"
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"skywork"
] | 086a56f1662ae157cc2723fa3cbf6624424af6400adbab9cc768cc5f35ad4cdf | [
{
"content": "Given positive integers \\(a\\) and \\(b\\) are each less than 10, find the smallest possible value for \\(2 \\cdot a - a \\cdot b\\).",
"role": "user"
}
] | Given positive integers \(a\) and \(b\) are each less than 10, find the smallest possible value for \(2 \cdot a - a \cdot b\). | "[\"-63\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1112, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15115"}] | {
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[
"16"
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"skywork"
] | 16ecc93796394afcc140aeddbf722e0a28deb19bd3d2c86adcaea6778388099d | [
{
"content": "17. In $\\triangle A B C$, $A B=A C=4$, point $P$ is any point on line segment $B C$, then $A P^{2}+P B \\cdot P C=$",
"role": "user"
}
] | 17. In $\triangle A B C$, $A B=A C=4$, point $P$ is any point on line segment $B C$, then $A P^{2}+P B \cdot P C=$ | "[\"16\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1099, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15116"}] | {
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[
"40"
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"skywork"
] | 278d6ca00179370a5cdade2f23aa9e48c24d8f681e920cc7d87bd70de8826397 | [
{
"content": "$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\\circ}$. Find the angular measure of arc $B D$.",
"role": "user"
}
] | $C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$. | "[\"40\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1103, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15117"}] | {
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[
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"skywork"
] | 2df84c6bea2fdc7e7db8726cc1973e178c3e2f643190be9d2f13de2dc9e2d0fc | [
{
"content": "5.082 If we expand all the brackets in the expression $(1+x)^{9}+(1+x)^{10}+\\ldots$ $\\ldots+(1+x)^{14}$ and combine like terms, we will get some polynomial. Determine the coefficient of $x^{9}$ in this polynomial without expanding the brackets.",
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}
] | 5.082 If we expand all the brackets in the expression $(1+x)^{9}+(1+x)^{10}+\ldots$ $\ldots+(1+x)^{14}$ and combine like terms, we will get some polynomial. Determine the coefficient of $x^{9}$ in this polynomial without expanding the brackets. | "[\"3003\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1097, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15118"}] | {
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[
"20100"
] | [
"skywork"
] | e936db36de7bf018646d9699cf6e52280556d71d5284df3fe39f9a854bc38946 | [
{
"content": "2. By how much is the sum of the squares of the first hundred even numbers greater than the sum of the squares of the first hundred odd numbers",
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}
] | 2. By how much is the sum of the squares of the first hundred even numbers greater than the sum of the squares of the first hundred odd numbers | "[\"20100\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1116, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15119"}] | {
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[
"1014049"
] | [
"skywork"
] | 6904b68e4318011897c98a64aba9a9242b7994bcd99d464f26627e431cabb2f2 | [
{
"content": "Compute the square of 1007 without a calculator.",
"role": "user"
}
] | Compute the square of 1007 without a calculator. | "[\"1014049\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1109, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15120"}] | {
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[
"10"
] | [
"skywork"
] | 662f5d9853c42b136ae1234e189a2b5cb1c76ebb4b9193930c4efb49eea61fc8 | [
{
"content": "Evaluate: $\\frac{10^{-2}5^0}{10^{-3}}$",
"role": "user"
}
] | Evaluate: $\frac{10^{-2}5^0}{10^{-3}}$ | "[\"10\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1129, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15121"}] | {
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[
"\\dfrac{1}{13}"
] | [
"skywork"
] | a09d6675b7fdd65f219a5ed8028d052878cc24dca354fc91d9ae7e88be99876d | [
{
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black... | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\... | "[\"\\\\dfrac{1}{13}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1107, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15122"}] | {
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[
"3029"
] | [
"skywork"
] | 522a880de62476069e4c8deb8f1e5a435046e8a2abb337edfc11504e8ae29b09 | [
{
"content": "Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.\n\n[i]Demetres Christofides, Cyprus[/i]",
"role": "user"
}
] | Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.
[i]Demetres Christofides, Cyprus[/i] | "[\" 3029 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1100, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15123"}] | {
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[
"37"
] | [
"skywork"
] | 588c74915348bd22edf96b9210825230a26990110b150d5016f6df31948d789d | [
{
"content": "Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, what is the smallest possible value of $a-b$?",
"role": "user"
}
] | Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, what is the smallest possible value of $a-b$? | "[\"37\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1128, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15124"}] | {
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[
"A(n)=[\\frac{\\sqrt{8n+1}-1}{2}]"
] | [
"skywork"
] | 39657e0b8338256c6a101133dc8bf1418d79a45b2de696299c1820a231060d12 | [
{
"content": "14. (1993 German Mathematical Olympiad) Every natural number $n$ greater than 2 can be expressed as the sum of several distinct natural numbers. Let the maximum number of these distinct natural numbers be $A(n)$. Find $A(n)$ (expressed in terms of $n$).",
"role": "user"
}
] | 14. (1993 German Mathematical Olympiad) Every natural number $n$ greater than 2 can be expressed as the sum of several distinct natural numbers. Let the maximum number of these distinct natural numbers be $A(n)$. Find $A(n)$ (expressed in terms of $n$). | "[\"A(n)=[\\\\frac{\\\\sqrt{8n+1}-1}{2}]\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1115, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15125"}] | {
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[
"-\\frac{8}{27}"
] | [
"skywork"
] | 40fb432c9986a9dc35bf8e3a575b979f679bb7efdad2aca7dacdc56520dd935b | [
{
"content": "Evaluate $x^2y^3z$ if $x = \\frac13$, $y = \\frac23$, and $z = -9$.",
"role": "user"
}
] | Evaluate $x^2y^3z$ if $x = \frac13$, $y = \frac23$, and $z = -9$. | "[\"-\\\\frac{8}{27}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1143, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15126"}] | {
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[
"391"
] | [
"skywork"
] | 8a25c98215677d2eeed1f001022f545b04863042ba681f007ce4b49b20b77fed | [
{
"content": "Given a cubic polynomial \\( q(x) \\) that satisfies \\( q(3) = 2 \\), \\( q(8) = 20 \\), \\( q(18) = 12 \\), and \\( q(25) = 32 \\). Find the sum \\( q(4) + q(5) + \\ldots + q(26) \\).",
"role": "user"
}
] | Given a cubic polynomial \( q(x) \) that satisfies \( q(3) = 2 \), \( q(8) = 20 \), \( q(18) = 12 \), and \( q(25) = 32 \). Find the sum \( q(4) + q(5) + \ldots + q(26) \). | "[\"391\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1102, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15127"}] | {
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[
"84"
] | [
"skywork"
] | 3ffb686278dd1530eca8b16cffdb8498d0236335b592e114442515df7057ea72 | [
{
"content": "One-half of one-seventh of $T$ equals one-third of one-fifth of 90. What is the value of $T$?",
"role": "user"
}
] | One-half of one-seventh of $T$ equals one-third of one-fifth of 90. What is the value of $T$? | "[\"84\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1119, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15128"}] | {
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[
"n = 6"
] | [
"skywork"
] | cd51f2774876cbd5d99db8915070b2e951f8840b7ae724670152db086087f7cd | [
{
"content": "Determine the smallest natural number $n$ having the following property: For every integer $p, p \\geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).",
"role": "user"
}
] | Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal). | "[\" n = 6 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1106, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15129"}, {"dataset": "skywork", "extra_inf... | {
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[
"0"
] | [
"skywork"
] | 5ea11ef72500fa447df7c47fa068f4c536370675b53d35336306edfdac9fbb81 | [
{
"content": "Let $a,b,c>0$ such that $a \\geq bc^2$ , $b \\geq ca^2$ and $c \\geq ab^2$ . Find the maximum value that the expression :\n$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.",
"role": "user"
}
] | Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression :
$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive. | "[\"0\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1110, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15130"}] | {
"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
} |
[
"150"
] | [
"skywork"
] | 5f6891b7f6a7fe79816b2429b2e2023fad49c16fdf5994f442e359c456ab85c9 | [
{
"content": "4. There are three sets of cards in red, yellow, and blue, each set containing five cards, marked with the letters $A, B, C, D, E$. If five cards are drawn from these 15 cards, with the requirement that the letters are all different and all three colors are included, then the number of different w... | 4. There are three sets of cards in red, yellow, and blue, each set containing five cards, marked with the letters $A, B, C, D, E$. If five cards are drawn from these 15 cards, with the requirement that the letters are all different and all three colors are included, then the number of different ways to draw the cards ... | "[\"150\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1101, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15131"}] | {
"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"
] | [
"skywork"
] | 1667957ec16df9374e3ad05efbc03b3800002a7448d169ad54843ee3edff9532 | [
{
"content": "If $a,b,c$ satisfy the system of equations \\begin{align*}b + c &= 12-3a \\\\\na+c &= -14 - 3b \\\\\na+b &= 7 - 3c,\n\\end{align*} what is $2a + 2b + 2c$?",
"role": "user"
}
] | If $a,b,c$ satisfy the system of equations \begin{align*}b + c &= 12-3a \\
a+c &= -14 - 3b \\
a+b &= 7 - 3c,
\end{align*} what is $2a + 2b + 2c$? | "[\"2\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1132, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15132"}] | {
"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
} |
[
"20\\mathrm{~}"
] | [
"skywork"
] | f6ce1e00bd3b2fb770f59658e4b2dacc03e709e5b3f32fff89c8cfd259f26676 | [
{
"content": "Jitka hiked a trail. After hiking $60 \\%$ of the length of the trail, she had $8 \\mathrm{~km}$ left to go. What is the length of the trail?\n(A) $28 \\mathrm{~km}$\n(B) $12.8 \\mathrm{~km}$\n(C) $11.2 \\mathrm{~km}$\n(D) $13 \\frac{1}{3} \\mathrm{~km}$\n(E) $20 \\mathrm{~km}$",
"role": "user... | Jitka hiked a trail. After hiking $60 \%$ of the length of the trail, she had $8 \mathrm{~km}$ left to go. What is the length of the trail?
(A) $28 \mathrm{~km}$
(B) $12.8 \mathrm{~km}$
(C) $11.2 \mathrm{~km}$
(D) $13 \frac{1}{3} \mathrm{~km}$
(E) $20 \mathrm{~km}$ | "[\"20\\\\mathrm{~}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1127, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 7, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15133"}] | {
"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
} |
[
"27"
] | [
"skywork"
] | 0fa7885267361cfdd1b3183d7b8b46ed8b09e73a7cca74ced40f1e78719400ae | [
{
"content": "How many fractions less than 1 are there, in which the numerator and the denominator are single-digit positive integers?",
"role": "user"
}
] | How many fractions less than 1 are there, in which the numerator and the denominator are single-digit positive integers? | "[\"27\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1122, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15134"}] | {
"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}{7}"
] | [
"skywork"
] | e0580176d9d5304d875aa0adbf6d31e47a654c811b28fe65bd76a1ddf813235e | [
{
"content": "The reciprocal of $\\frac{3}{10}$ is $\\left(\\frac{1}{x}+1\\right)$. What is the value of $x$ ?\n(A) $\\frac{7}{3}$\n(B) $\\frac{3}{13}$\n(C) $\\frac{3}{7}$\n(D) $\\frac{5}{3}$\n(E) $\\frac{3}{5}$",
"role": "user"
}
] | The reciprocal of $\frac{3}{10}$ is $\left(\frac{1}{x}+1\right)$. What is the value of $x$ ?
(A) $\frac{7}{3}$
(B) $\frac{3}{13}$
(C) $\frac{3}{7}$
(D) $\frac{5}{3}$
(E) $\frac{3}{5}$ | "[\"\\\\frac{3}{7}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1136, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15135"}] | {
"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
} |
[
"144"
] | [
"skywork"
] | b6ef7509a9227feb170ad043045909420dd07fc1722db27de91bcceda6c0b6cd | [
{
"content": "$$\n\\begin{array}{l}\n\\left(a_{1}+a_{2}+a_{3}\\right)\\left(b_{1}+b_{2}+b_{3}+b_{4}\\right) \\\\\n\\cdot\\left(c_{1}+c_{2}+c_{3}+c_{4}+c_{5}+c_{6}\\right)\\left(d_{1}+d_{2}\\right)\n\\end{array}\n$$\n\nHow many different terms are there in the expanded form of the polynomial product?",
"role... | $$
\begin{array}{l}
\left(a_{1}+a_{2}+a_{3}\right)\left(b_{1}+b_{2}+b_{3}+b_{4}\right) \\
\cdot\left(c_{1}+c_{2}+c_{3}+c_{4}+c_{5}+c_{6}\right)\left(d_{1}+d_{2}\right)
\end{array}
$$
How many different terms are there in the expanded form of the polynomial product? | "[\"144\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1114, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15136"}] | {
"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"
] | [
"skywork"
] | 24b3d892ed65a27f1b7ea2f778c8f6bb3c55d404520d013ac837e3ef9a3b383f | [
{
"content": "8. (10 points) Player A and Player B are having a table tennis match, with a best-of-three format. In each game, the first to score 11 points and the opponent scoring less than 10 points wins; if the score is tied at 10, the first to score 2 more points wins. The total points scored by both A and ... | 8. (10 points) Player A and Player B are having a table tennis match, with a best-of-three format. In each game, the first to score 11 points and the opponent scoring less than 10 points wins; if the score is tied at 10, the first to score 2 more points wins. The total points scored by both A and B are 30 points. Witho... | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1104, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15137"}] | {
"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
} |
[
"13,7,4"
] | [
"skywork"
] | fe37fbb90070e5a10b386e44ff517f6ccdaab8fb02de1c1293cf5e121928599a | [
{
"content": "Example 16: Three piles of chess pieces, A, B, and C, are moved as follows: First, take chess pieces from pile A and place them into piles B and C according to the number of pieces in piles B and C; then, take chess pieces from pile B and place them into piles A and C according to the number of pi... | Example 16: Three piles of chess pieces, A, B, and C, are moved as follows: First, take chess pieces from pile A and place them into piles B and C according to the number of pieces in piles B and C; then, take chess pieces from pile B and place them into piles A and C according to the number of pieces in piles A and C;... | "[\"13,7,4\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1124, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15138"}] | {
"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"
] | [
"skywork"
] | da769d9a75c82f229e5d411d5d1a52ee379b0e1ef2ed782ec24db247f6eb07f7 | [
{
"content": "If\n\\[\\frac{y}{x-z}=\\frac{x+y}{z}=\\frac{x}{y}\\]\nfor three positive numbers $x,y$ and $z$, all different, then $\\frac{x}{y}=$\n$\\text{(A) } \\frac{1}{2}\\quad \\text{(B) } \\frac{3}{5}\\quad \\text{(C) } \\frac{2}{3}\\quad \\text{(D) } \\frac{5}{3}\\quad \\text{(E) } 2$",
"role": "user"... | If
\[\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}\]
for three positive numbers $x,y$ and $z$, all different, then $\frac{x}{y}=$
$\text{(A) } \frac{1}{2}\quad \text{(B) } \frac{3}{5}\quad \text{(C) } \frac{2}{3}\quad \text{(D) } \frac{5}{3}\quad \text{(E) } 2$ | "[\"2\"]" | train-math-numinamath1.5_amc_aime | [{"dataset": "skywork", "extra_info": {"index": 1138, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15139"}] | {
"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
} |
[
"\\sqrt{2}"
] | [
"skywork"
] | 0a48e93a1ed622a4e2dbf1ef0e2bb47cff28cc579aa8e11d8055ce5476b61ebc | [
{
"content": "Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \\in \\mathbb{R}$). \n(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$. \n(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the sh... | Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \in \mathbb{R}$).
(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$.
(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the shortest? And what is the sh... | "[\"\\\\sqrt{2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1117, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15140"}] | {
"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
} |
[
"2/15"
] | [
"skywork"
] | ef61243fe353f8dc7f8241d6964aadd62cfafa6bdd2dadacff5a08ad4e34b10f | [
{
"content": "Let the function \\( f(x) = \\left| \\log_{2} x \\right| \\). Real numbers \\( a \\) and \\( b \\) (where \\( a < b \\)) satisfy the following conditions:\n\\[\n\\begin{array}{l}\nf(a+1) = f(b+2), \\\\\nf(10a + 6b + 22) = 4\n\\end{array}\n\\]\n\nFind \\( ab \\).",
"role": "user"
}
] | Let the function \( f(x) = \left| \log_{2} x \right| \). Real numbers \( a \) and \( b \) (where \( a < b \)) satisfy the following conditions:
\[
\begin{array}{l}
f(a+1) = f(b+2), \\
f(10a + 6b + 22) = 4
\end{array}
\]
Find \( ab \). | "[\"2/15\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1113, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15141"}] | {
"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
} |
[
"175"
] | [
"skywork"
] | af5b7124bf0c403b48286280665d0f280811f1b4a22578b404e178edecc70b42 | [
{
"content": "3 Let $X=\\{0,1,2, \\cdots, 9\\}, F=\\left\\{A_{1}, A_{2}, \\cdots, A_{k}\\right\\}$ where each element $A_{i}$ is a non-empty subset of $X$, and for any $1 \\leqslant i<j \\leqslant k$, we have $\\left|A_{i} \\cap A_{j}\\right| \\leqslant 2$, find the maximum value of $k$. (No.",
"role": "use... | 3 Let $X=\{0,1,2, \cdots, 9\}, F=\left\{A_{1}, A_{2}, \cdots, A_{k}\right\}$ where each element $A_{i}$ is a non-empty subset of $X$, and for any $1 \leqslant i<j \leqslant k$, we have $\left|A_{i} \cap A_{j}\right| \leqslant 2$, find the maximum value of $k$. (No. | "[\"175\"]" | train-math-numinamath1.5_inequalities | [{"dataset": "skywork", "extra_info": {"index": 1134, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15142"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"2584"
] | [
"skywork"
] | 038333698c6257ea1c4f2a255e58d596224aac7f0fc0db6038562bbd8edc605b | [
{
"content": "Determine integers $a$ and $b$ such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$.",
"role": "user"
}
] | Determine integers $a$ and $b$ such that $x^2 - x - 1$ is a factor of $ax^{18} + bx^{17} + 1$. | "[\"2584\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1150, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15143"}] | {
"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
} |
[
"50"
] | [
"skywork"
] | d3eec37f761ec6d585fd7f7d56e418ac54d68f374a3a502856ac5c9604826a0e | [
{
"content": "Question 240, Set $S$ satisfies the following conditions: (1) The elements in $S$ are all positive integers not exceeding 100; (2) For any $a, b \\in S, a \\neq b$, there exists a $c \\in S$ different from $a, b$, such that $(a+b, c)=1$; (3) For any $a, b \\in S, a \\neq b$, there exists a $c \\in... | Question 240, Set $S$ satisfies the following conditions: (1) The elements in $S$ are all positive integers not exceeding 100; (2) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)=1$; (3) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, s... | "[\"50\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1118, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15144"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"(8,2,3),(7,4,2),(6,6,1)"
] | [
"skywork"
] | d8452c332d9396765fffb8793f17d660fc028b314f0f96f7e4b85b204dbbd827 | [
{
"content": "184. In the apartment, there are 13 people, cats, and flies. Together, they have 42 legs, with each fly having 6 legs. How many people, cats, and flies were there separately? List all possible answers.",
"role": "user"
}
] | 184. In the apartment, there are 13 people, cats, and flies. Together, they have 42 legs, with each fly having 6 legs. How many people, cats, and flies were there separately? List all possible answers. | "[\"(8,2,3),(7,4,2),(6,6,1)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1120, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15145"}] | {
"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
} |
[
"27"
] | [
"skywork"
] | 3b6d043f3057f37064e3f0fd88443f8c90d202d35783a3ef6a55b42f52e59f89 | [
{
"content": "A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green be... | A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads in this necklace? ... | "[\"27\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1146, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15147"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"12"
] | [
"skywork"
] | 1e518dd8616ad1e094e4b5b46198ccbb484b7174c858f589bdfe254608657f9e | [
{
"content": "I4.3 Given that there are $R$ odd numbers in the digits of the product of the two $Q$-digit numbers 1111...11 and $9999 \\ldots 99$, find the value of $R$.",
"role": "user"
}
] | I4.3 Given that there are $R$ odd numbers in the digits of the product of the two $Q$-digit numbers 1111...11 and $9999 \ldots 99$, find the value of $R$. | "[\"12\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1126, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15148"}] | {
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[
"9"
] | [
"skywork"
] | f36245d9afa3ac3f74ae634ed6fadf3abd650e559e0ef32ea8ec046c2975796a | [
{
"content": "Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution.",
"role": "user"
}
] | Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | "[\"9\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1151, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15149"}] | {
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} |
[
"x_{1}=\\frac{7}{4},x_{2}=\\frac{7}{3},x_{3}=\\frac{7}{2}"
] | [
"skywork"
] | e2bb7971108d124b93db3865ccd13f05f16a7a7e03448748fcd4c52598166d69 | [
{
"content": "## 3. Solve the equation in the set of real numbers\n\n$$\nx\\lfloor x\\rfloor-5 x+7=0 \\text {, }\n$$\n\nwhere $\\lfloor x\\rfloor$ denotes the floor function of $x$, i.e., the greatest integer $k$ such that $k \\leqq x$. (For example, $\\lfloor\\sqrt{2}\\rfloor=1 \\quad \\text{and} \\quad \\lflo... | ## 3. Solve the equation in the set of real numbers
$$
x\lfloor x\rfloor-5 x+7=0 \text {, }
$$
where $\lfloor x\rfloor$ denotes the floor function of $x$, i.e., the greatest integer $k$ such that $k \leqq x$. (For example, $\lfloor\sqrt{2}\rfloor=1 \quad \text{and} \quad \lfloor-3.1\rfloor=-4$). | "[\"x_{1}=\\\\frac{7}{4},x_{2}=\\\\frac{7}{3},x_{3}=\\\\frac{7}{2}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1123, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15150"}] | {
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[
"a>0 \\text{ or } a<-1"
] | [
"skywork"
] | a9ff11ec369423ecc99454021ec13f5489b4206d08d6f14adb3e2cf114725b1e | [
{
"content": "6. For real numbers $u, v$, define an operation “ * ” as: $u * v = uv + v$. If the equation $x * (a * x) = -\\frac{1}{4}$ has two distinct real roots, then the range of real numbers $a$ that satisfy the condition is . $\\qquad$",
"role": "user"
}
] | 6. For real numbers $u, v$, define an operation “ * ” as: $u * v = uv + v$. If the equation $x * (a * x) = -\frac{1}{4}$ has two distinct real roots, then the range of real numbers $a$ that satisfy the condition is . $\qquad$ | "[\"a>0 \\\\text{ or } a<-1\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1111, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15151"}] | {
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[
"22"
] | [
"skywork"
] | 44cb7db4f77c3ecf00f4b886eae5853f3b945d37a3e6d6d75bee236fab9adeca | [
{
"content": "Given that $15^{-1} \\equiv 31 \\pmod{53}$, find $38^{-1} \\pmod{53}$, as a residue modulo 53.",
"role": "user"
}
] | Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53. | "[\"22\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1141, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15152"}] | {
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} |
[
"[-\\frac{1}{2},1)"
] | [
"skywork"
] | a8d831bc4657f2e121df6373ae0b8e93b3e36177859cabd7ccf8b7c21ff04062 | [
{
"content": "Example 7 The range of the function $y=\\frac{x^{2}-1}{x^{2}+2}$ is $\\qquad$ .",
"role": "user"
}
] | Example 7 The range of the function $y=\frac{x^{2}-1}{x^{2}+2}$ is $\qquad$ . | "[\"[-\\\\frac{1}{2},1)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1145, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15153"}] | {
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} |
[
"\\frac{3}{128}"
] | [
"skywork"
] | e0061c808bbbabba60627b9e1f36685c17bfec0f7b874550eb001f7f09888d90 | [
{
"content": "A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads?",
"role": "user"
}
] | A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads? | "[\"\\\\frac{3}{128}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1154, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15154"}, {"dataset": "skywork", "extra_... | {
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"reference_components": 1,
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} |
[
"100"
] | [
"skywork"
] | 2989dcd31530c306dcfdfc5521617316d6ff0cddb305e0e99521224fb39af484 | [
{
"content": "Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \\cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ .",
"role": "user"
}
] | Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ . | "[\"100\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 1149, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15155"}, {"dataset": "skywork", "extra_inf... | {
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} |
[
"8"
] | [
"skywork"
] | e6193accc574f5f3e9035bbce027b7ccb2d1d690da391ace4d7d7cafef131c1f | [
{
"content": "In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$.",
"role": "user"
}
] | In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$. | "[\"8\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1140, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15156"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
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} |
[
"f(n) = n"
] | [
"skywork"
] | 014331c2f6d72076d06a1ec58f2ec495c5bb6708df38a2eeaa31e8a81d6175c2 | [
{
"content": "Find all functions $f : \\mathbb{N} \\to \\mathbb{N}$ such that for any positive integers $m, n$ the number\n$$(f(m))^2+ 2mf(n) + f(n^2)$$\nis the square of an integer.\n\n[i]Proposed by Fedir Yudin[/i]",
"role": "user"
}
] | Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number
$$(f(m))^2+ 2mf(n) + f(n^2)$$
is the square of an integer.
[i]Proposed by Fedir Yudin[/i] | "[\" f(n) = n \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1131, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15157"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"50"
] | [
"skywork"
] | 133d01f884acc9d24808e400bbffe517ca07f4ef8386b6147e90638bf3d3f280 | [
{
"content": "There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar?",
"role": "user"
}
] | There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar? | "[\"50\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1121, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15158"}] | {
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"reference_components": 1,
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} |
[
"0.75"
] | [
"skywork"
] | 0a90f4b6a9e547fe0fd8c8fc779b8bfe6f64cad6383cd409892ff8beb94815dc | [
{
"content": "At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 1... | At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when sh... | "[\"0.75\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1133, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 15, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15159"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"D"
] | [
"skywork"
] | 4c07df3abd4896467b158e55c06aff1338abd1d7ee3d20a26b2965ea160033f9 | [
{
"content": "Let $ n\\left(A\\right)$ be the number of distinct real solutions of the equation $ x^{6} \\minus{}2x^{4} \\plus{}x^{2} \\equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\\left(A\\right)$ is\n\n$\\textbf{(A)}\\ \\left\\{0,1,2,3,4,5,6\\right\\} \\\\ \\textbf{(B)}\\ ... | Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is
$\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ ... | "[\"D\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1125, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15160"}] | {
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[
"12504"
] | [
"skywork"
] | 1453ed38a275b0a41fdfd079e429fe027e6c9c408f474c4d503e6ff9e0e50539 | [
{
"content": "How many five-digit numbers divisible by 3 are there that include the digit 6?",
"role": "user"
}
] | How many five-digit numbers divisible by 3 are there that include the digit 6? | "[\"12504\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1155, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15161"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{n(n-1)}{4}"
] | [
"skywork"
] | 2673df380532504d19a5c19297aa200ed0d6b9c988abcaa04ab8a9846f4d45f6 | [
{
"content": "27.9. (New York, 81). The sequence ( $i_{1} ; i_{2} ; \\ldots ; i_{n}$ ) is composed of the first $n$ natural numbers arranged in a random order. Find the average number of inversions (disorders) in this sequence, where an inversion is defined as each pair of numbers $i_{j}>i_{k}$, for which $j<k$... | 27.9. (New York, 81). The sequence ( $i_{1} ; i_{2} ; \ldots ; i_{n}$ ) is composed of the first $n$ natural numbers arranged in a random order. Find the average number of inversions (disorders) in this sequence, where an inversion is defined as each pair of numbers $i_{j}>i_{k}$, for which $j<k$. | "[\"\\\\frac{n(n-1)}{4}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1153, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15162"}] | {
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[
"9,11,25"
] | [
"skywork"
] | 0d2dda587ddb3d907b4a327c774668ab55e44ed360d120ffd1879a6fd34aaad4 | [
{
"content": "$\\left.\\begin{array}{l}{[\\text { Decimal numeral system }]} \\\\ {\\left[\\begin{array}{l}\\text { Case analysis }\\end{array}\\right]} \\\\ {[\\underline{\\text { Divisibility rules for 11 }}]} \\\\ {[\\underline{\\text { Divisibility rules for 3 and 9 }}]}\\end{array}\\right]$\n\nA two-digit ... | $\left.\begin{array}{l}{[\text { Decimal numeral system }]} \\ {\left[\begin{array}{l}\text { Case analysis }\end{array}\right]} \\ {[\underline{\text { Divisibility rules for 11 }}]} \\ {[\underline{\text { Divisibility rules for 3 and 9 }}]}\end{array}\right]$
A two-digit number is appended to the right of a certain... | "[\"9,11,25\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1139, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15163"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2\\sqrt{29}"
] | [
"skywork"
] | fc33765548c0ed64f8f0f915fac87a37b1a33507098a24c19a0cee67fc1d13f1 | [
{
"content": "Let $\\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\\neq A$ denote the point on $\\odot (ABC)$ for which $AQ\\perp BC$. The circumcircle of $\\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are... | Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and ... | "[\"2\\\\sqrt{29}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1135, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15164"}] | {
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} |
[
"n = p^{2010}"
] | [
"skywork"
] | 0c34bc6df259870d63c7162dbb1f36de333cba1ebc80690894d3b198b807ba94 | [
{
"content": "Find all positive integers $n$ such that the equation $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \\leq y$.",
"role": "user"
}
] | Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$. | "[\" n = p^{2010} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1152, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15165"}] | {
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} |
[
"340"
] | [
"skywork"
] | 24363d150577cca5e4716c1d5ad6690de37aa16b6747f4a0329eca3f38970249 | [
{
"content": "The increasing sequence of positive integers $a_1, a_2, a_3, \\dots$ is defined by the rule\n\\[a_{n + 2} = a_{n + 1} + a_n\\]\nfor all $n \\ge 1.$ If $a_7 = 210$, then find $a_8.$",
"role": "user"
}
] | The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$ | "[\"340\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1144, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15166"}] | {
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[
"0"
] | [
"skywork"
] | 3e69edaedf0188a88b7565b619c8b8710ebdcf3b480d6dc0e374013733b3905a | [
{
"content": "[ Absolute value ]\n\nSolve the equation $|x-2|+|x-1|+|x|+|x+1|+|x+2|=6$.\n\n#",
"role": "user"
}
] | [ Absolute value ]
Solve the equation $|x-2|+|x-1|+|x|+|x+1|+|x+2|=6$.
# | "[\"0\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1161, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15167"}] | {
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[
"\\text{a sphere centered at } P \\text{ with radius } R"
] | [
"skywork"
] | 0d4f9146a8c6bed06dee27bf2c045326b9f9b3a600e5c80cc48aafacc0fe423f | [
{
"content": "We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as ed... | We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus ... | "[\"\\\\text{a sphere centered at } P \\\\text{ with radius } R\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1137, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15168"}] | {
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[
"[1,\\frac{1+\\sqrt{5}}{2})"
] | [
"skywork"
] | e2d2cf55e4f313dda3242adb3cad57e30cf266daf250e01b324beb14e5620920 | [
{
"content": "8. Let $0<\\alpha \\leqslant \\beta \\leqslant \\gamma$, and $\\alpha+\\beta+\\gamma=\\pi$, then the range of $\\min \\left\\{\\frac{\\sin \\beta}{\\sin \\alpha}, \\frac{\\sin \\gamma}{\\sin \\beta}\\right\\}$ is",
"role": "user"
}
] | 8. Let $0<\alpha \leqslant \beta \leqslant \gamma$, and $\alpha+\beta+\gamma=\pi$, then the range of $\min \left\{\frac{\sin \beta}{\sin \alpha}, \frac{\sin \gamma}{\sin \beta}\right\}$ is | "[\"[1,\\\\frac{1+\\\\sqrt{5}}{2})\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1156, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15169"}] | {
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} |
[
"25"
] | [
"skywork"
] | 48e7640f5af8ec4b5bf741f0529cbfd61a66d8a5f3eb60d40b6361d2f70761bd | [
{
"content": "If $x=5$ and $y=x+3$ and $z=3 y+1$, the value of $z$ is\n(A) 7\n(B) 25\n(C) 12\n(D) 46\n(E) 19",
"role": "user"
}
] | If $x=5$ and $y=x+3$ and $z=3 y+1$, the value of $z$ is
(A) 7
(B) 25
(C) 12
(D) 46
(E) 19 | "[\"25\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1169, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15170"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1782"
] | [
"skywork"
] | d3e94aad20c665dce675a1390efb09e6f06878743401ce0e2f2caeaa8234dd1d | [
{
"content": "A sequence of numbers is arranged in the following pattern: \\(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \\cdots\\). Starting from the first number on the left, find the sum of the first 99 numbers.",
"role": "user"
}
] | A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers. | "[\"1782\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1158, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15171"}] | {
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"label_correctness_independently_verified": false,
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"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{4}"
] | [
"skywork"
] | d9b9e4ddc6701420e1b91ffba113a5cb4fdb7d874a1ff8935c43bb4e3355a24d | [
{
"content": "In a dice game, a player rolls two dice. His score is the larger of the two numbers on the dice. For example, if he rolls 3 and 5 , his score is 5 , and if he rolls 4 and 4 , his score is 4 . What is the probability that his score is 3 or less?\n(A) $\\frac{1}{4}$\n(B) $\\frac{7}{36}$\n(C) $\\frac... | In a dice game, a player rolls two dice. His score is the larger of the two numbers on the dice. For example, if he rolls 3 and 5 , his score is 5 , and if he rolls 4 and 4 , his score is 4 . What is the probability that his score is 3 or less?
(A) $\frac{1}{4}$
(B) $\frac{7}{36}$
(C) $\frac{5}{36}$
(D) $\frac{1}{3}$
(... | "[\"\\\\frac{1}{4}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1167, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15172"}] | {
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[
"98751"
] | [
"skywork"
] | c49b45019300f7369921edde424559c7b0b6370fc6cabf2d9368d86eb3653e0f | [
{
"content": "Task 3. Determine the largest five-digit number such that the product of its digits is 2520?",
"role": "user"
}
] | Task 3. Determine the largest five-digit number such that the product of its digits is 2520? | "[\"98751\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1172, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15173"}] | {
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"reference_components": 1,
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} |
[
"(3,-5)"
] | [
"skywork"
] | d5a428f1637c49f14ebdb3b674eb179a077325d771f81203abeca1f424f6b103 | [
{
"content": "[Coordinate method on the plane $]$ $[\\underline{\\text { Orthocenter and orthotriangle }]}$\n\nGiven points $A(5, -1), B(4, -8), C(-4, -4)$. Find the coordinates of the intersection point of the altitudes of triangle $ABC$.\n\n#",
"role": "user"
}
] | [Coordinate method on the plane $]$ $[\underline{\text { Orthocenter and orthotriangle }]}$
Given points $A(5, -1), B(4, -8), C(-4, -4)$. Find the coordinates of the intersection point of the altitudes of triangle $ABC$.
# | "[\"(3,-5)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1159, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15174"}] | {
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} |
[
"\\sqrt{5}"
] | [
"skywork"
] | 559ca9d0a2d497bdac81fca118659b2527b5047996bdc5b605741bfe70478751 | [
{
"content": "Let $P$ be a point on the line\n\\[\\begin{pmatrix} 3 \\\\ -1 \\\\ 2 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -2 \\\\ 1 \\end{pmatrix}\\]and let $Q$ be a point on the line\n\\[\\begin{pmatrix} 0 \\\\ 0 \\\\ 4 \\end{pmatrix} + s \\begin{pmatrix} 1 \\\\ 2 \\\\ -1 \\end{pmatrix}.\\]Find the shortes... | Let $P$ be a point on the line
\[\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\]and let $Q$ be a point on the line
\[\begin{pmatrix} 0 \\ 0 \\ 4 \end{pmatrix} + s \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}.\]Find the shortest possible distance $PQ.$ | "[\"\\\\sqrt{5}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1170, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15175"}] | {
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} |
[
"345"
] | [
"skywork"
] | d9206775c1838daf90265f2477588144b38c102feedcc2d2acb4f91a6d3a851f | [
{
"content": "Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included.",
"role": "user"
}
] | Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included. | "[\"345\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1165, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15176"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{\\sqrt{2}}{2}"
] | [
"skywork"
] | e12abfd1cc4d152c0b68bc45357bffeac94c6da663395f5ae1b9ebb700dbccdf | [
{
"content": "2. Let $x, y, z$ be real numbers, not all zero. Then the maximum value of the function $f(x, y, z)=\\frac{x y+y z}{x^{2}+y^{2}+z^{2}}$ is $\\qquad$",
"role": "user"
}
] | 2. Let $x, y, z$ be real numbers, not all zero. Then the maximum value of the function $f(x, y, z)=\frac{x y+y z}{x^{2}+y^{2}+z^{2}}$ is $\qquad$ | "[\"\\\\frac{\\\\sqrt{2}}{2}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1164, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15177"}] | {
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"plain_numeric_reference": false,
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} |
[
"CADBE"
] | [
"skywork"
] | 7e3c7284a79659dc88b946dbcc63599ac328887af72c55798ddfb172b68ea586 | [
{
"content": "20. (6 points) Five people, A, B, C, D, and E, are guessing the top five individual subject total scores in the class:\nA: “The first place is $D$, the fifth place is $E$. ”\nB: “The second place is $A$, the fourth place is $C$.”\nC: “The third place is $D$, the fourth place is $A$ ”,\nD: “The fir... | 20. (6 points) Five people, A, B, C, D, and E, are guessing the top five individual subject total scores in the class:
A: “The first place is $D$, the fifth place is $E$. ”
B: “The second place is $A$, the fourth place is $C$.”
C: “The third place is $D$, the fourth place is $A$ ”,
D: “The first place is $C$, the third... | "[\"CADBE\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1160, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15178"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"skywork"
] | d73ff130de9f0c328f433f7bc4343a610b40acc28e547fbaa898279f1f4d7667 | [
{
"content": "G2.2 If $(0.0025 \\times 40)^{b}=\\frac{1}{100}$, find $b$.",
"role": "user"
}
] | G2.2 If $(0.0025 \times 40)^{b}=\frac{1}{100}$, find $b$. | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1174, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15179"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"6"
] | [
"skywork"
] | 1910b112c3b1c8f3ad8ac621e575ba4465adb82ae5ecde1401dd516394e3ad0e | [
{
"content": "Pedro decided to take all his children, boys and girls, for ice cream at the Ice Cream Math parlor. At the parlor, there are 12 different ice cream flavors, and each child ordered a combo with 3 scoops of ice cream. After leaving the parlor, Pedro realized that exactly two scoops of each available... | Pedro decided to take all his children, boys and girls, for ice cream at the Ice Cream Math parlor. At the parlor, there are 12 different ice cream flavors, and each child ordered a combo with 3 scoops of ice cream. After leaving the parlor, Pedro realized that exactly two scoops of each available flavor were ordered i... | "[\"6\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1147, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15180"}] | {
"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
} |
[
"16"
] | [
"skywork"
] | af1535b4949b18f5f902e6d9fdca363bda23172499e5ed513671bd298fccc374 | [
{
"content": "The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated.",
"role": "user"
}
] | The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated. | "[\"16\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1171, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15181"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"21"
] | [
"skywork"
] | ec3de8a9588bf50ccc576d530047a9a49fb3d2d717eb63673e22c69cf85d1ede | [
{
"content": "Three, let the three sides of $\\triangle A B C$ be $a, b, c$, and all are numbers. If it satisfies $\\angle A=3 \\angle B$, try to find the minimum value of the perimeter and provide a proof.",
"role": "user"
}
] | Three, let the three sides of $\triangle A B C$ be $a, b, c$, and all are numbers. If it satisfies $\angle A=3 \angle B$, try to find the minimum value of the perimeter and provide a proof. | "[\"21\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1148, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15182"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{2}"
] | [
"skywork"
] | aa5f483af5f47444324f37a2db33423e313dd10d533922987474775c285173ef | [
{
"content": "Let $S$ be the set of all real values of $x$ with $0 < x < \\pi$ such that $\\sin x$, $\\cos x$, and $\\tan x$ (with $\\tan x$ defined) form the side lengths (in some order) of a right triangle. Compute the sum of $\\tan^2 x$ over all $x$ in $S$.",
"role": "user"
}
] | Let $S$ be the set of all real values of $x$ with $0 < x < \pi$ such that $\sin x$, $\cos x$, and $\tan x$ (with $\tan x$ defined) form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$. | "[\"\\\\sqrt{2}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1162, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15183"}] | {
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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}{N+1}"
] | [
"skywork"
] | e8d1b5172bfa0743bf16a0c396b5742f1f518475154cd85a1c06fd5241a6bcd8 | [
{
"content": "Compute \\[\\frac{(N-1)!(N)}{(N+1)!}.\\]",
"role": "user"
}
] | Compute \[\frac{(N-1)!(N)}{(N+1)!}.\] | "[\"\\\\frac{1}{N+1}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1185, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15184"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"128"
] | [
"skywork"
] | 5a353f404fac03aae22696aa8515a1cc2cb19de23949ee3ff986b1b1f23563c2 | [
{
"content": "Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at mos... | Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters ... | "[\"128\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1168, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15185"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"13"
] | [
"skywork"
] | cf7dbaae16aa8b7ea47b2c81c32897a9fdc02f64117271e27ffa5281be89c92b | [
{
"content": "Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.",
"role": "user"
}
] | Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes. | "[\"13\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1187, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15186"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{546}{5}"
] | [
"skywork"
] | 528a87ef687fd137c652c014f19a9fc9b7ec9612b9cd060ee2a623516740b39f | [
{
"content": "Solve for $x$: $\\sqrt[3]{20x + \\sqrt[3]{20x + 13}} = 13.$",
"role": "user"
}
] | Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$ | "[\"\\\\frac{546}{5}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1201, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15187"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
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} |
[
"184320"
] | [
"skywork"
] | a838b4506e55c48b6e9d976b090a19aedc8f04a40eb12000af3bb565c61f9b1e | [
{
"content": "For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.",
"role": "user"
}
] | For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products. | "[\"184320\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1200, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15188"}] | {
"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
} |
[
"250"
] | [
"skywork"
] | 766aad8aa9d5410c0b41711c7a5c5d16fa65dbc63b9e9f7cf1ccd173d4bad08a | [
{
"content": "Euler's formula states that for a convex polyhedron with $V\\,$ vertices, $E\\,$ edges, and $F\\,$ faces, $V-E+F=2\\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\\,$ vertices, $T\\,$ triangular faces and $P^{}_{}$ pentagonal f... | Euler's formula states that for a convex polyhedron with $V\,$ vertices, $E\,$ edges, and $F\,$ faces, $V-E+F=2\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\,$ vertices, $T\,$ triangular faces and $P^{}_{}$ pentagonal faces meet. What is the value... | "[\"250\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 1157, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15189"}] | {
"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
} |
[
"20"
] | [
"skywork"
] | 04b78f4ad109ec1526e315a96f1eec6ede2354dcb4aca40aaed8a052d89f34a2 | [
{
"content": "19. (6 points) There is a strip of paper, on which there are three types of markings, dividing the strip into 6 equal parts, 10 equal parts, and 12 equal parts along its length. Now, if we cut the strip along all the markings with scissors, the strip will be divided into $\\qquad$ parts.",
"ro... | 19. (6 points) There is a strip of paper, on which there are three types of markings, dividing the strip into 6 equal parts, 10 equal parts, and 12 equal parts along its length. Now, if we cut the strip along all the markings with scissors, the strip will be divided into $\qquad$ parts. | "[\"20\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1178, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15190"}] | {
"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
} |
[
"9\\pi"
] | [
"skywork"
] | f7db1a8733fdf031259795ed34e5b118a7ab7c65222f8760aed30c9252a7244c | [
{
"content": "Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.",
"role": "user"
}
] | Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments. | "[\"9\\\\pi\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1177, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15191"}] | {
"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{27}{5}"
] | [
"skywork"
] | eaa5eaa2aafd03970bd3c826043333a7c1bc575fc82b5831d28022b50ca061bb | [
{
"content": "2. If $(a b):(a c):(b c)=5: 3: 1$, calculate the value of the expression $\\frac{a^{5} b^{2}}{c^{3}}: \\frac{a^{3} b^{5}}{c^{4}}$.",
"role": "user"
}
] | 2. If $(a b):(a c):(b c)=5: 3: 1$, calculate the value of the expression $\frac{a^{5} b^{2}}{c^{3}}: \frac{a^{3} b^{5}}{c^{4}}$. | "[\"\\\\frac{27}{5}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1175, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15192"}] | {
"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"
] | [
"skywork"
] | e98b24c0afcb4774d91308dd4233b61bd768747a1c16b307b0c6e4d2448ddbc2 | [
{
"content": "1. Given that $a$, $b$, and $c$ are constants, and for any real number $x$ we have\n$$\nx^{3}+2 x+c=(x+1)\\left(x^{2}+a x+b\\right) \\text {. }\n$$\n\nthen $a b c=$ . $\\qquad$",
"role": "user"
}
] | 1. Given that $a$, $b$, and $c$ are constants, and for any real number $x$ we have
$$
x^{3}+2 x+c=(x+1)\left(x^{2}+a x+b\right) \text {. }
$$
then $a b c=$ . $\qquad$ | "[\"-9\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1179, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15193"}] | {
"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
} |
[
"4983"
] | [
"skywork"
] | 1ed24746ae0906b437dfa5e62bfd9f5ffa5c2d5860aa41eac0564492b42ea018 | [
{
"content": "Given a sequence of positive integers $a_1, a_2, a_3, \\dots, a_{n}$, define the \\emph{power tower function} \\[f(a_1, a_2, a_3, \\dots, a_{n})=a_1^{a_2^{a_3^{\\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\\] Let $b_1, b_2, b_3, \\dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 201... | Given a sequence of positive integers $a_1, a_2, a_3, \dots, a_{n}$, define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let $b_1, b_2, b_3, \dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 2017 inclusive, \[f(a_1, a_2, a_... | "[\"4983\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1176, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15194"}] | {
"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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