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 |
|---|---|---|---|---|---|---|---|---|
[
"14"
] | [
"skywork"
] | 1df55e20195941b757d57525c675c35902b803d8562bfcf853f1cb7520dcc013 | [
{
"content": "14.28 The sequence $\\left\\{a_{n}\\right\\}$ is defined as $a_{1}=1, \\quad a_{n+1}=a_{n}+\\frac{1}{a_{n}} \\quad(n \\geqslant 1)$. Find $\\left[a_{100}\\right]$.\n(Japan Mathematical Olympiad, 1990)",
"role": "user"
}
] | 14.28 The sequence $\left\{a_{n}\right\}$ is defined as $a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n}} \quad(n \geqslant 1)$. Find $\left[a_{100}\right]$.
(Japan Mathematical Olympiad, 1990) | "[\"14\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2911, "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:16840"}] | {
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} |
[
"(1)"
] | [
"skywork"
] | 04598ba623d688d5f4beeab1f2ecfb4be14c1100ba4ce5e2f9e7f76d57eca4d1 | [
{
"content": "Among the following propositions, the true one is numbered \\_\\_\\_\\_\\_\\_. \n(1) The negation of the proposition \"For all $x>0$, $x^2-x\\leq0$\" is \"There exists an $x>0$ such that $x^2-x>0$.\" \n(2) If $A>B$, then $\\sin A > \\sin B$. \n(3) Given a sequence $\\{a_n\\}$, \"The sequence $a... | Among the following propositions, the true one is numbered \_\_\_\_\_\_.
(1) The negation of the proposition "For all $x>0$, $x^2-x\leq0$" is "There exists an $x>0$ such that $x^2-x>0$."
(2) If $A>B$, then $\sin A > \sin B$.
(3) Given a sequence $\{a_n\}$, "The sequence $a_n, a_{n+1}, a_{n+2}$ forms a geometric s... | "[\"(1)\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2881, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16841"}] | {
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} |
[
"8"
] | [
"skywork"
] | 9c9b0cd565b0e099573e92a39aec16198feb081e370c892eb8a36f1b2b93f654 | [
{
"content": "For how many integers $n$, does the equation system \\[\\begin{array}{rcl}\n2x+3y &=& 7\\\\\n5x + ny &=& n^2\n\\end{array}\\] have a solution over integers?\n\n$ \n\\textbf{(A)}\\ 0\n\\qquad\\textbf{(B)}\\ 3\n\\qquad\\textbf{(C)}\\ 4\n\\qquad\\textbf{(D)}\\ 8\n\\qquad\\textbf{(E)}\\ \\text{None of... | For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | "[\"8\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2900, "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:16842"}] | {
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"any_reference_component_parses": true,
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"reference_components": 1,
"reference_present": true
} |
[
"1346"
] | [
"skywork"
] | 390dcd4318f68eadddce81b00993bf83987c3ed3d817ec0f9f0e9e19a9c3470f | [
{
"content": "On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, \"Now we too can form a corr... | On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, "Now we too can form a correct circle." How many k... | "[\"1346\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2884, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16843"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-\\frac{4}{3}"
] | [
"skywork"
] | 7020748f4ebec207ec86d7c9f220eae6e232e829ba745b0ce0a5377b80999529 | [
{
"content": "Example 1 Given the sets $P=\\left\\{m^{2}-4, m+1,-3\\right\\}, Q=\\{m-3,2 m-1,3 m+1\\}$, if $P \\cap Q=$ $\\{-3\\}$, find the value of the real number $m$.\n\n untranslated text remains in its original format and line breaks are preserved.",
"role": "user"
}
] | Example 1 Given the sets $P=\left\{m^{2}-4, m+1,-3\right\}, Q=\{m-3,2 m-1,3 m+1\}$, if $P \cap Q=$ $\{-3\}$, find the value of the real number $m$.
untranslated text remains in its original format and line breaks are preserved. | "[\"-\\\\frac{4}{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2896, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16844"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"4037"
] | [
"skywork"
] | 6042c1b8cfd20e08a7b58a32c4779cef67a8075c172ef287a41c576f24a3b4a8 | [
{
"content": "Exercise 14. Consider a grid of size $2019 \\times 2019$. On this grid, pebbles are placed. A configuration is said to be beautiful if there does not exist a parallelogram formed by four pebbles $A B C D$, such that $A, B, C$, and $D$ are not all aligned.\n\nWhat is the maximum number of pebbles t... | Exercise 14. Consider a grid of size $2019 \times 2019$. On this grid, pebbles are placed. A configuration is said to be beautiful if there does not exist a parallelogram formed by four pebbles $A B C D$, such that $A, B, C$, and $D$ are not all aligned.
What is the maximum number of pebbles that can be placed on the ... | "[\"4037\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2901, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 10, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16845"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"n - 2 + \\sqrt{2}"
] | [
"skywork"
] | 100f77a7e2b4c8bae2dd8a8aaa9c6adfc617df260ad41216442237f91d21638b | [
{
"content": "Let $ n \\geq 3$ be an odd integer. Determine the maximum value of\r\n\\[ \\sqrt{|x_{1}\\minus{}x_{2}|}\\plus{}\\sqrt{|x_{2}\\minus{}x_{3}|}\\plus{}\\ldots\\plus{}\\sqrt{|x_{n\\minus{}1}\\minus{}x_{n}|}\\plus{}\\sqrt{|x_{n}\\minus{}x_{1}|},\\]\r\nwhere $ x_{i}$ are positive real numbers from the i... | Let $ n \geq 3$ be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where $ x_{i}$ are positive real numbers from the interval $ [0,1]$. | "[\"n - 2 + \\\\sqrt{2}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2894, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16846"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"16"
] | [
"skywork"
] | 5f733ecc1b87c25deb1f83e6671b668b259fe4d23f0970c3192448b4efee3e56 | [
{
"content": "13.186. A brigade of masons took on the task of laying $432 \\mathrm{~m}^{3}$ of masonry, but in reality, 4 fewer people showed up for work. How many masons are there in the brigade if it is known that each working mason had to lay $9 \\mathrm{~m}^{3}$ more than initially planned?",
"role": "u... | 13.186. A brigade of masons took on the task of laying $432 \mathrm{~m}^{3}$ of masonry, but in reality, 4 fewer people showed up for work. How many masons are there in the brigade if it is known that each working mason had to lay $9 \mathrm{~m}^{3}$ more than initially planned? | "[\"16\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2885, "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:16847"}] | {
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"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0"
] | [
"skywork"
] | 169a7365ce48d7950a641925fca0adc7aa4ed5c60ca57f421ac88c81f0861cee | [
{
"content": "2. Given $a, b, c$ are distinct. Find the value of $\\frac{2 a-b-c}{(a-b)(a-c)}$ $+\\frac{2 b-c-a}{(b-c)(b-a)}+\\frac{2 c-a-b}{(c-a)(c-b)}$.",
"role": "user"
}
] | 2. Given $a, b, c$ are distinct. Find the value of $\frac{2 a-b-c}{(a-b)(a-c)}$ $+\frac{2 b-c-a}{(b-c)(b-a)}+\frac{2 c-a-b}{(c-a)(c-b)}$. | "[\"0\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2907, "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:16848"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"(-2,0),(-3,0),(0,-2),(-1,2)"
] | [
"skywork"
] | 2957df99cf6dd5d3c5f77fe0c557070d94e32b34c752d0219abbb873cba8d400 | [
{
"content": "Find all the integer solutions of the equation\n\n$$\n9 x^{2} y^{2}+9 x y^{2}+6 x^{2} y+18 x y+x^{2}+2 y^{2}+5 x+7 y+6=0\n$$",
"role": "user"
}
] | Find all the integer solutions of the equation
$$
9 x^{2} y^{2}+9 x y^{2}+6 x^{2} y+18 x y+x^{2}+2 y^{2}+5 x+7 y+6=0
$$ | "[\"(-2,0),(-3,0),(0,-2),(-1,2)\"]" | train-math-numinamath1.5_olympiads_ref | [{"dataset": "skywork", "extra_info": {"index": 2908, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16849"}] | {
"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
} |
[
"8065"
] | [
"skywork"
] | 3c0190f1eb2832b23995a289f33193dde76a03656ca0e14a589a2822984c6e3f | [
{
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=4$, $a_3=9$, $a_n=a_{n-1}+a_{n-2}-a_{n-3}$, for $n=4,5,...$, calculate $a_{2017}$.",
"role": "user"
}
] | Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=4$, $a_3=9$, $a_n=a_{n-1}+a_{n-2}-a_{n-3}$, for $n=4,5,...$, calculate $a_{2017}$. | "[\"8065\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2903, "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:16850"}] | {
"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
} |
[
"9:36"
] | [
"skywork"
] | 15753d1d2ae91816e1987305c0b91bc37b063628732c991abe799044afd56242 | [
{
"content": "At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time?",
"role": "user"
}
] | At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time? | "[\"9:36\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2904, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16851"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2021"
] | [
"skywork"
] | d72cb939f5a9e5cf3511297e3e790c6a038bbae85f7a4d5c63f60f34f2a70acb | [
{
"content": "1. On the board, three quadratic equations are written:\n\n$$\n\\begin{gathered}\n2020 x^{2}+b x+2021=0 \\\\\n2019 x^{2}+b x+2020=0 \\\\\nx^{2}+b x+2019=0\n\\end{gathered}\n$$\n\nFind the product of the roots of all the equations written on the board, given that each of them has two real roots.",
... | 1. On the board, three quadratic equations are written:
$$
\begin{gathered}
2020 x^{2}+b x+2021=0 \\
2019 x^{2}+b x+2020=0 \\
x^{2}+b x+2019=0
\end{gathered}
$$
Find the product of the roots of all the equations written on the board, given that each of them has two real roots. | "[\"2021\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2883, "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:16852"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"66"
] | [
"skywork"
] | 62585f03a0bc56ca6a85a2460b09ab56c800ce74aeaf246bfde18aeaca8b3820 | [
{
"content": "Given several rectangular prisms with edge lengths of $2, 3,$ and $5$, aligned in the same direction to form a cube with an edge length of $90$, how many small rectangular prisms does one diagonal of the cube intersect?",
"role": "user"
}
] | Given several rectangular prisms with edge lengths of $2, 3,$ and $5$, aligned in the same direction to form a cube with an edge length of $90$, how many small rectangular prisms does one diagonal of the cube intersect? | "[\"66\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2902, "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:16853"}] | {
"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
} |
[
"14"
] | [
"skywork"
] | 6f295fce15b9c98798e969dce41875e3ce7fd14df4249651de715ca055556cad | [
{
"content": "What is the sum of all real numbers \\(x\\) for which \\(|x^2 - 14x + 45| = 3?\\)\nA) 12\nB) 14\nC) 16\nD) 18",
"role": "user"
}
] | What is the sum of all real numbers \(x\) for which \(|x^2 - 14x + 45| = 3?\)
A) 12
B) 14
C) 16
D) 18 | "[\"14\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2914, "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:16854"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"4"
] | [
"skywork"
] | 84eef4f7eae17ca7f0a2867b6c3103b165a614c62f77b692cbad40d24eb26946 | [
{
"content": "6. In the right-angled triangle $ABC$ with the right angle at vertex $B$, on the longer leg $BC$, a segment $BD$ equal to the shorter leg is laid off from point $B$. Points $K$ and $N$ are the feet of the perpendiculars dropped from points $B$ and $D$ to the hypotenuse, respectively. Find the leng... | 6. In the right-angled triangle $ABC$ with the right angle at vertex $B$, on the longer leg $BC$, a segment $BD$ equal to the shorter leg is laid off from point $B$. Points $K$ and $N$ are the feet of the perpendiculars dropped from points $B$ and $D$ to the hypotenuse, respectively. Find the length of segment $BK$, if... | "[\"4\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2905, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16856"}] | {
"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 {\\pi}{3}"
] | [
"skywork"
] | 79d805cee07993b54d6d7a51d4248af2490060d76183041099c957575adc02ef | [
{
"content": "If $$\\sin\\alpha= \\frac {4}{7} \\sqrt {3}$$ and $$\\cos(\\alpha+\\beta)=- \\frac {11}{14}$$, and $\\alpha$, $\\beta$ are acute angles, then $\\beta= \\_\\_\\_\\_\\_\\_$.",
"role": "user"
}
] | If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | "[\"\\\\frac {\\\\pi}{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2910, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16858"}] | {
"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
} |
[
"19"
] | [
"skywork"
] | 5790e5e25f07f8000b5d72e2effaa02efbfe7f6d421d05ad707b687c58378b9e | [
{
"content": "Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \\ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\\frac{1}{r_i}+\\frac{1}{r_j}-\\frac{1}{r_ir_j}$ where $i < j$ and $i, j \\in \\{1, ... , 10\\}$. If $P(0) = Q(1) = 2$, then $\\log_2 (|P(1)|)$ can be wri... | Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively p... | "[\"19\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2909, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "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:16860"}] | {
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[
"\\frac{13 + 8\\sqrt{2}}{41}"
] | [
"skywork"
] | 7c94151c8f4881b81b489e2f22372cc58926f91c5d01c22b1039bfdedb2b9a17 | [
{
"content": "In isosceles right triangle $ ABC$ a circle is inscribed. Let $ CD$ be the hypotenuse height ($ D\\in AB$), and let $ P$ be the intersection of inscribed circle and height $ CD$. In which ratio does the circle divide segment $ AP$?",
"role": "user"
}
] | In isosceles right triangle $ ABC$ a circle is inscribed. Let $ CD$ be the hypotenuse height ($ D\in AB$), and let $ P$ be the intersection of inscribed circle and height $ CD$. In which ratio does the circle divide segment $ AP$? | "[\" \\\\frac{13 + 8\\\\sqrt{2}}{41} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2916, "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:16861"}] | {
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} |
[
"55"
] | [
"skywork"
] | 3efd8335bec19ae7d9ed97764179f902ab100993e225677c560dd915804b36d7 | [
{
"content": "Problem 6.3. Three merchants: Foma, Yerema, and Julius met in Novgorod. If Foma gives Yerema 70 gold coins, then Yerema and Julius will have the same amount of money. If Foma gives Yerema 40 gold coins, then Foma and Julius will have the same amount of money. How many gold coins should Foma give Y... | Problem 6.3. Three merchants: Foma, Yerema, and Julius met in Novgorod. If Foma gives Yerema 70 gold coins, then Yerema and Julius will have the same amount of money. If Foma gives Yerema 40 gold coins, then Foma and Julius will have the same amount of money. How many gold coins should Foma give Yerema so that they bot... | "[\"55\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2918, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "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:16862"}] | {
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} |
[
"49"
] | [
"skywork"
] | ea00b6f4923fbbcd5c444b6e0b54d3266a5b1ce4734737fbf1ffe47dd68383b3 | [
{
"content": "6. Among the 27 points consisting of the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube, the number of groups of three collinear points is\n(A) 57\n(B) 49\n(C) 43\n(D) 37",
"role": "user"
}
] | 6. Among the 27 points consisting of the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube, the number of groups of three collinear points is
(A) 57
(B) 49
(C) 43
(D) 37 | "[\"49\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2924, "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:16863"}] | {
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"reference_components": 1,
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[
"960"
] | [
"skywork"
] | 3b27fdf77c64f199c525430bd5efcf6ea49dbe5107dc8771982ebafdc663fb89 | [
{
"content": "Two classmates, A and B, live in the same neighborhood and leave the neighborhood gate at the same time to go to school. Initially, A travels at a speed of 40 meters per minute, while B travels at a speed of 60 meters per minute. After A has walked half the distance, they realize they will be late... | Two classmates, A and B, live in the same neighborhood and leave the neighborhood gate at the same time to go to school. Initially, A travels at a speed of 40 meters per minute, while B travels at a speed of 60 meters per minute. After A has walked half the distance, they realize they will be late at this pace and incr... | "[\"960\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2906, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "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:16864"}] | {
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"reference_components": 1,
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} |
[
"\\frac{8}{27}"
] | [
"skywork"
] | 56dd95dd55d4a80482d017bf368e4af2f85a44edf65597cbeb90c3dde9d86b89 | [
{
"content": "Let \\( x, y, z, w \\in [0,1] \\). Find the maximum value of \\( S = x^2 y + y^2 z + z^2 w + w^2 x - xy^2 - yz^2 - zw^2 - wx^2 \\).",
"role": "user"
}
] | Let \( x, y, z, w \in [0,1] \). Find the maximum value of \( S = x^2 y + y^2 z + z^2 w + w^2 x - xy^2 - yz^2 - zw^2 - wx^2 \). | "[\"\\\\frac{8}{27}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2923, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16865"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"90"
] | [
"skywork"
] | 2a8b2e0fb663236d0db9c08d73148e0182da233bc9c8ecd5cf52951b8e4b2a6c | [
{
"content": "An $8 \\times 8 \\times n$ rectangular prism is made up from $1 \\times 1 \\times 1$ cubes. Suppose that $A$ is the surface area of the prism and $B$ is the combined surface area of the $1 \\times 1 \\times 1$ cubes that make up the prism. What is the sum of the values of $n$ for which $\\frac{B}{... | An $8 \times 8 \times n$ rectangular prism is made up from $1 \times 1 \times 1$ cubes. Suppose that $A$ is the surface area of the prism and $B$ is the combined surface area of the $1 \times 1 \times 1$ cubes that make up the prism. What is the sum of the values of $n$ for which $\frac{B}{A}$ is an integer?
(A) 86
(B)... | "[\"90\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2929, "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:16866"}] | {
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[
"101"
] | [
"skywork"
] | 97c8f53702239c100c92f6575863f44cf82aac94bc03324abd5871610aa5288b | [
{
"content": "16. Let $n$ be a positive integer. When $n$ is divided by 902,802 and 702 , the remainders are 602,502 and 402 respectively. Find the minimum remainder when $n$ is divided by 2005 .\n(2 marks)設 $n$ 為正整數。當 $n$ 除以 902 、 802 及 702 時, 其餘數分別是 602 、 502 及 402 。求 $n$ 除以 2005 時的餘數的最小值。",
"role": "user... | 16. Let $n$ be a positive integer. When $n$ is divided by 902,802 and 702 , the remainders are 602,502 and 402 respectively. Find the minimum remainder when $n$ is divided by 2005 .
(2 marks)設 $n$ 為正整數。當 $n$ 除以 902 、 802 及 702 時, 其餘數分別是 602 、 502 及 402 。求 $n$ 除以 2005 時的餘數的最小值。 | "[\"101\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2919, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 10, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16867"}] | {
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} |
[
"217"
] | [
"skywork"
] | 7b703631452a4764f480c5ce17b9de35cec32e6e22bbed5720340e446f5e2184 | [
{
"content": "The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by 37?",
"role": "user"
}
] | The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by 37? | "[\"217\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 2939, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "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:16868"}, {"dataset": "skywork", "extra_inf... | {
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"reference_components": 1,
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} |
[
"7"
] | [
"skywork"
] | 2974496ae61876520f6479416d7872a2045576667490097e62e69312bcdaae92 | [
{
"content": "1. (mathematics) There are scales with two pans, 4 weights of 2 kg each, 3 weights of 3 kg each, and two weights of 5 kg each. In how many different ways can a 12 kg load be balanced on the scales, if the weights are allowed to be placed on both pans?",
"role": "user"
}
] | 1. (mathematics) There are scales with two pans, 4 weights of 2 kg each, 3 weights of 3 kg each, and two weights of 5 kg each. In how many different ways can a 12 kg load be balanced on the scales, if the weights are allowed to be placed on both pans? | "[\"7\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2921, "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:16869"}] | {
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} |
[
"(-1, -1, -1), (0, 0, 0), (1, 1, 1)"
] | [
"skywork"
] | 41406ae89ec0b671befde288b0df458b487c2f59e655a15f32e3d5e57e36608c | [
{
"content": "Find all real numbers $a, b, c$ that satisfy\n$$ 2a - b =a^2b, \\qquad 2b-c = b^2 c, \\qquad 2c-a= c^2 a.$$",
"role": "user"
}
] | Find all real numbers $a, b, c$ that satisfy
$$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$ | "[\" (-1, -1, -1), (0, 0, 0), (1, 1, 1) \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2945, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16870"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{32}"
] | [
"skywork"
] | 80f37f14d96ce82358846c7c94e51e8a4ecb106f354ac43475c8835a96be001d | [
{
"content": "A square-based frustum has a base edge and every side edge of 4. The edge of its top face is 2. What is the maximum distance between two vertices of the frustum?",
"role": "user"
}
] | A square-based frustum has a base edge and every side edge of 4. The edge of its top face is 2. What is the maximum distance between two vertices of the frustum? | "[\"\\\\sqrt{32}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2956, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16871"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"1297"
] | [
"skywork"
] | 06abce78a33edf9250f2819e6b845d8a448ec12886b8d6a3a7f67c69469f036a | [
{
"content": "6. Given that $a, b, c$ are distinct positive integers. If the set\n$$\n\\{a+b, b+c, c+a\\}=\\left\\{n^{2},(n+1)^{2},(n+2)^{2}\\right\\} \\text {, }\n$$\n\nwhere $n \\in \\mathbf{Z}_{+}$. Then the minimum value of $a^{2}+b^{2}+c^{2}$ is",
"role": "user"
}
] | 6. Given that $a, b, c$ are distinct positive integers. If the set
$$
\{a+b, b+c, c+a\}=\left\{n^{2},(n+1)^{2},(n+2)^{2}\right\} \text {, }
$$
where $n \in \mathbf{Z}_{+}$. Then the minimum value of $a^{2}+b^{2}+c^{2}$ is | "[\"1297\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2933, "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:16872"}] | {
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} |
[
"\\cos66,\\cos78,\\cos138,\\cos150"
] | [
"skywork"
] | 651ff605eaae658f5b66e156aa721dfff42ba12e2810052ed186a69aaa9a5a23 | [
{
"content": "5. It is known that the number $\\cos 6^{0}$ is a root of the equation $32 t^{5}-40 t^{3}+10 t-\\sqrt{3}=0$. Find the other four roots of this equation. (Answers in the problem should be compact expressions, not containing summation signs, ellipses, and radicals.)\n\n#",
"role": "user"
}
] | 5. It is known that the number $\cos 6^{0}$ is a root of the equation $32 t^{5}-40 t^{3}+10 t-\sqrt{3}=0$. Find the other four roots of this equation. (Answers in the problem should be compact expressions, not containing summation signs, ellipses, and radicals.)
# | "[\"\\\\cos66,\\\\cos78,\\\\cos138,\\\\cos150\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2932, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16873"}, {"dataset": "skywork", "extra_i... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"1116"
] | [
"skywork"
] | e9b078e7a347eb1a741cb54c138d1096feaa515238b4fcdf7f0c47a9edd557be | [
{
"content": "Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins al... | Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is ... | "[\"1116\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2934, "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:16874"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{17}{13}"
] | [
"skywork"
] | 3fcecae069ac0f581fa09459739f663e2b443afd494a0bebe4ce4839709854ea | [
{
"content": "Given the hyperbola $C$: $\\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$.\n(I) Find the range of the eccentricity $e$ of the hyperbola $C$.\n(II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\\overrightarrow{PA} ... | Given the hyperbola $C$: $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$.
(I) Find the range of the eccentricity $e$ of the hyperbola $C$.
(II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\overrightarrow{PA} = \frac{5}{12}\overrightar... | "[\"\\\\frac{17}{13}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2913, "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:16875"}] | {
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} |
[
"\\left(a^{2}-b^{2}-c\\right)^{2}+(2 a b-d)^{2}=4\\left(a^{2}+b^{2}\\right)"
] | [
"skywork"
] | 30347db4c3f1a166f6e5e082f0ecc10688fce34b1fde3eb84ce46da37cf56075 | [
{
"content": "4・154 From the system of equations\n$$\\left\\{\\begin{array}{l}\na=\\cos u+\\cos v+\\cos w \\\\\nb=\\sin u+\\sin v+\\sin w \\\\\nc=\\cos 2 u+\\cos 2 v+\\cos w \\\\\nd=\\sin 2 u+\\sin 2 v+\\sin 2 w\n\\end{array}\\right.$$\n\neliminate \\( u, v, w \\).",
"role": "user"
}
] | 4・154 From the system of equations
$$\left\{\begin{array}{l}
a=\cos u+\cos v+\cos w \\
b=\sin u+\sin v+\sin w \\
c=\cos 2 u+\cos 2 v+\cos w \\
d=\sin 2 u+\sin 2 v+\sin 2 w
\end{array}\right.$$
eliminate \( u, v, w \). | "[\"\\\\left(a^{2}-b^{2}-c\\\\right)^{2}+(2 a b-d)^{2}=4\\\\left(a^{2}+b^{2}\\\\right)\"]" | train-math-numinamath1.5_inequalities | [{"dataset": "skywork", "extra_info": {"index": 2928, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16876"}] | {
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[
"2"
] | [
"skywork"
] | 76fed3464f8acd308f18ee0f6877dea4d10b2c0c3b75e414dc4b20b5859265a5 | [
{
"content": "* In $\\triangle A B C$, the sides opposite to $\\angle A, \\angle B, \\angle C$ are $a, b, c$ respectively. If $a, b, c$ form an arithmetic sequence, and $c=10, a \\cos A=b \\cos B, A \\neq B$, then the inradius of $\\triangle A B C$ is equal to",
"role": "user"
}
] | * In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. If $a, b, c$ form an arithmetic sequence, and $c=10, a \cos A=b \cos B, A \neq B$, then the inradius of $\triangle A B C$ is equal to | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2926, "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:16877"}] | {
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} |
[
"x = \\frac{1}{2}, y < \\frac{2 - 3\\sqrt[3]{2}}{4}"
] | [
"skywork"
] | 089d12d44e5909b0940a6a99b4c1b11650ade2753b0bbb3b95b0d3c5c834f3ab | [
{
"content": "[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.\n[b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.\n\n[i](I. Gorodnin)[/i]",
... | [b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
[b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
[i](I. Gorodnin)[/i] | "[\" x = \\\\frac{1}{2}, y < \\\\frac{2 - 3\\\\sqrt[3]{2}}{4} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2927, "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:16878"}] | {
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[
"1.5"
] | [
"skywork"
] | fbd2a667b3c2215eb89b81afa66d602311010ebce76423dae55401ff6594dfe7 | [
{
"content": "In rectangle \\(A B C D\\), point \\(E\\) is the midpoint of side \\(C D\\). On side \\(B C\\), point \\(F\\) is chosen such that \\(\\angle A E F\\) is a right angle. Find the length of segment \\(F C\\) if \\(A F = 7\\) and \\(B F = 4\\).",
"role": "user"
}
] | In rectangle \(A B C D\), point \(E\) is the midpoint of side \(C D\). On side \(B C\), point \(F\) is chosen such that \(\angle A E F\) is a right angle. Find the length of segment \(F C\) if \(A F = 7\) and \(B F = 4\). | "[\"1.5\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2936, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "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:16879"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"805"
] | [
"skywork"
] | 40f360c27c8efa262784b035a32f6dac3eeed8e7d0fdb73db8295540a2c9a081 | [
{
"content": "$$\n\\begin{array}{l}\n\\text { 8. Let } \\\\\n\\begin{array}{l}\nA=\\{1,2, \\cdots, 2014\\}, \\\\\nB_{i}=\\left\\{x_{i}, y_{i}\\right\\}(i=1,2, \\cdots, t)\n\\end{array}\n\\end{array}\n$$\n\nbe $t$ pairwise disjoint binary subsets of $A$, and satisfy the conditions\n$$\n\\begin{array}{l}\nx_{i}+y... | $$
\begin{array}{l}
\text { 8. Let } \\
\begin{array}{l}
A=\{1,2, \cdots, 2014\}, \\
B_{i}=\left\{x_{i}, y_{i}\right\}(i=1,2, \cdots, t)
\end{array}
\end{array}
$$
be $t$ pairwise disjoint binary subsets of $A$, and satisfy the conditions
$$
\begin{array}{l}
x_{i}+y_{i} \leqslant 2014(i=1,2, \cdots, t), \\
x_{i}+y_{i}... | "[\"805\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2931, "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:16880"}] | {
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"label_correctness_independently_verified": false,
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"reference_present": true
} |
[
"2"
] | [
"skywork"
] | c65a9ea44eca9ef410d78e30d6af1620d110825c8ffe840039ac1376117ae788 | [
{
"content": "Example 6 In a rectangle $A B C D$ with an area of 1 (including the boundary) there are 5 points, among which no three points are collinear. Find the minimum number of triangles, with these 5 points as vertices, whose area is not greater than $\\frac{1}{4}$.",
"role": "user"
}
] | Example 6 In a rectangle $A B C D$ with an area of 1 (including the boundary) there are 5 points, among which no three points are collinear. Find the minimum number of triangles, with these 5 points as vertices, whose area is not greater than $\frac{1}{4}$. | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2944, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16881"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"4"
] | [
"skywork"
] | 7cbb9e5340f02d3b498b87a046c944c7024c93813ed9e56cac3a945e18843ac1 | [
{
"content": "8. In $\\triangle A B C$, the three sides $a, b, c$ form an arithmetic sequence. Find the value of $5 \\cos A-4 \\cos A \\cos C+5 \\cos C$.",
"role": "user"
}
] | 8. In $\triangle A B C$, the three sides $a, b, c$ form an arithmetic sequence. Find the value of $5 \cos A-4 \cos A \cos C+5 \cos C$. | "[\"4\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2943, "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:16882"}, {"dataset": "skywork", "extra_inf... | {
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"reference_components": 1,
"reference_present": true
} |
[
"13"
] | [
"skywork"
] | 78dcae5c19039c6674ade7a7f439101b11fe94cc1254a8333778170d5d4afd14 | [
{
"content": "Fix a sequence $a_1,a_2,a_3\\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.",
"role": "user"
}
] | Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$. | "[\"13\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2917, "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:16883"}, {"dataset": "skywork", "extra_... | {
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[
"10"
] | [
"skywork"
] | 17a878ad3a0f35e1dae18d8d96daccc555da5c4f91b4371dad43d62b3cbd8c81 | [
{
"content": "A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?",
"role": "user"
}
] | A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? | "[\"10\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2925, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16884"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2 \\sqrt {3}}{3}"
] | [
"skywork"
] | 9755f637f420720fea8dc1c08143a678583232a60f23df34a6d6328ef0ecaf74 | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\\sin 2A+\\sin (A-B)=\\sin C$, and $A\\neq \\frac{\\pi}{2}$.\n- (I) Find the value of $\\frac{a}{b}$;\n- (II) If $c=2$ and $C= \\frac{\\pi}{3}$, find the area of $\\triangle A... | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\sin 2A+\sin (A-B)=\sin C$, and $A\neq \frac{\pi}{2}$.
- (I) Find the value of $\frac{a}{b}$;
- (II) If $c=2$ and $C= \frac{\pi}{3}$, find the area of $\triangle ABC$. | "[\"\\\\frac{2 \\\\sqrt {3}}{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2937, "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:16885"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
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} |
[
"10"
] | [
"skywork"
] | 7c153393afe333103cad69dbe15780d18f5fcd57499942d823e7383b419a5ad9 | [
{
"content": "For which value of $m$, there is no triple of integer $(x,y,z)$ such that $3x^2+4y^2-5z^2=m$?\n\n$ \\textbf{(A)}\\ 16\n\\qquad\\textbf{(B)}\\ 14\n\\qquad\\textbf{(C)}\\ 12\n\\qquad\\textbf{(D)}\\ 10\n\\qquad\\textbf{(E)}\\ 8\n$",
"role": "user"
}
] | For which value of $m$, there is no triple of integer $(x,y,z)$ such that $3x^2+4y^2-5z^2=m$?
$ \textbf{(A)}\ 16
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 8
$ | "[\"10\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2958, "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:16886"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"31"
] | [
"skywork"
] | 062f32abe2909fa1ff0e73211d437cd7e8119958df3a0db6c4db14536e18b4cf | [
{
"content": "Problem 6. Thirty-three bogatyrs (Russian knights) were hired to guard Lake Luka for 240 coins. The cunning Chernomor can divide the bogatyrs into squads of any size (or record all in one squad), and then distribute the entire salary among the squads. Each squad divides its coins equally, and the ... | Problem 6. Thirty-three bogatyrs (Russian knights) were hired to guard Lake Luka for 240 coins. The cunning Chernomor can divide the bogatyrs into squads of any size (or record all in one squad), and then distribute the entire salary among the squads. Each squad divides its coins equally, and the remainder goes to Cher... | "[\"31\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2946, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16887"}] | {
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"reference_present": true
} |
[
"124"
] | [
"skywork"
] | 26bbc8b8ac80146fd218c6c46a0cbdd7feb953b09f93fb55e7e8acd7e9cbf96a | [
{
"content": "Given the length of four sides of an inscribed convex octagon is $2$, and the length of the other four sides is $6\\sqrt{2}$, calculate the area of this octagon.",
"role": "user"
}
] | Given the length of four sides of an inscribed convex octagon is $2$, and the length of the other four sides is $6\sqrt{2}$, calculate the area of this octagon. | "[\"124\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2951, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "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:16889"}] | {
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} |
[
"n \\text{ must be odd}"
] | [
"skywork"
] | 0303e6fce75a08ef7175f552ac62122d8c60da826b383e8d43651182bdbe41b5 | [
{
"content": "Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the ... | Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectiv... | "[\"n \\\\text{ must be odd}\"]" | train-math-omnimath | [{"dataset": "skywork", "extra_info": {"index": 2922, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16891"}] | {
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} |
[
"k = 2"
] | [
"skywork"
] | e6468c53636dd8840f93e991fcc96d35fdd766b77b35bc1d0685910b106ea3db | [
{
"content": "16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the fina... | 16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it ha... | "[\" k = 2 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2920, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16892"}] | {
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} |
[
"a + 1 + \\left\\lfloor k - a \\right\\rfloor"
] | [
"skywork"
] | 29e96f2031a96aae65568c704ee21923f6d2a3334568939ee46ebb0adaece1e0 | [
{
"content": "Let $k\\geqslant 2$ be an integer. Consider the sequence $(x_n)_{n\\geqslant 1}$ defined by $x_1=a>0$ and $x_{n+1}=x_n+\\lfloor k/x_n\\rfloor$ for $n\\geqslant 1.$ Prove that the sequence is convergent and determine its limit.",
"role": "user"
}
] | Let $k\geqslant 2$ be an integer. Consider the sequence $(x_n)_{n\geqslant 1}$ defined by $x_1=a>0$ and $x_{n+1}=x_n+\lfloor k/x_n\rfloor$ for $n\geqslant 1.$ Prove that the sequence is convergent and determine its limit. | "[\" a + 1 + \\\\left\\\\lfloor k - a \\\\right\\\\rfloor \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2935, "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:16893"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"skywork"
] | 834ac7ce882e01416b7f156fbd614b6047fca91dd773222a82520e5f2b33dc11 | [
{
"content": "1. (1993 National High School Mathematics Competition) What are the last two digits of the integer $\\left[\\frac{10^{93}}{10^{31}+3}\\right]$? (Write the tens digit first, then the units digit)",
"role": "user"
}
] | 1. (1993 National High School Mathematics Competition) What are the last two digits of the integer $\left[\frac{10^{93}}{10^{31}+3}\right]$? (Write the tens digit first, then the units digit) | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2960, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "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:16894"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"skywork"
] | 8a9517a33f2232288a3baa3ea9d92c2873069e3c248f4b6e9dc61ebf743f1f8a | [
{
"content": "What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ?",
"role": "user"
}
] | What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ? | "[\"3\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2966, "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:16895"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{\\sqrt{5}}{2}"
] | [
"skywork"
] | c36c8487e3efe12b1f5bbb4885bb0ca653f93e907cbf79e927b5708cf4880c6a | [
{
"content": "Any six points are taken inside or on a rectangle with dimensions $1 \\times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.",
"r... | Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | "[\"\\\\frac{\\\\sqrt{5}}{2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2949, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16896"}] | {
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[
"b = 2^k"
] | [
"skywork"
] | 993301d3f6682eeac0fff000348ffc44f6a5d0cd29f2a78a4b6eb913d1da4b8d | [
{
"content": "Let $n$ and $k$ be integers, $1\\le k\\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:\n(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.\n(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.\n(iii) For all distin... | Let $n$ and $k$ be integers, $1\le k\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:
(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.
(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.
(iii) For all distinct $a,a'$ in $A$, $a$ does ... | "[\" b = 2^k \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2938, "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:16897"}] | {
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} |
[
"99972"
] | [
"skywork"
] | 61480ebaf20bb4396070d6237848ec903cf1106b6c9ce88e0f2ae14e18b88952 | [
{
"content": "The \ffive-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest \ffive-digit number which is divisible by the sum of its digits",
"role": "user"
}
] | The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits | "[\"99972\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2957, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "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:16898"}, {"dataset": "skywork", "extra_in... | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1154"
] | [
"skywork"
] | 77bc40c0a9a2541b4aac01ac9a959c25f06cb756751f94077e72d0f6390e57cf | [
{
"content": "1. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set\n$$\n\\begin{array}{l}\nA=\\{y \\mid y=[x]+[2 x]+[4 x], x \\in \\mathbf{R}\\}, \\\\\nB=\\{1,2, \\cdots, 2019\\} .\n\\end{array}\n$$\n\nThen the number of elements in $A \\cap B$ is $\\qquad$",
"role": "user"
}
] | 1. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set
$$
\begin{array}{l}
A=\{y \mid y=[x]+[2 x]+[4 x], x \in \mathbf{R}\}, \\
B=\{1,2, \cdots, 2019\} .
\end{array}
$$
Then the number of elements in $A \cap B$ is $\qquad$ | "[\"1154\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2953, "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:16899"}] | {
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} |
[
"38"
] | [
"skywork"
] | 4f706315069acaa9b4255da6d95583379c471e1d23348a0f0510f85be2f501c9 | [
{
"content": "What is the largest even integer that cannot be written as the sum of two odd composite numbers?",
"role": "user"
}
] | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | "[\"38\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2968, "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:16900"}, {"dataset": "skywork", "extra_inf... | {
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} |
[
"e - 1"
] | [
"skywork"
] | 7c40e271e14bb9bc4092fc5703ecac62bca4f014dc6b5cad2daf63119a02b44c | [
{
"content": "Let $x_0,x_1,x_2,\\dots$ be the sequence such that $x_0=1$ and for $n\\ge 0,$\n\\[x_{n+1}=\\ln(e^{x_n}-x_n)\\]\n(as usual, the function $\\ln$ is the natural logarithm). Show that the infinite series\n\\[x_0+x_1+x_2+\\cdots\\]\nconverges and find its sum.",
"role": "user"
}
] | Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$
\[x_{n+1}=\ln(e^{x_n}-x_n)\]
(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series
\[x_0+x_1+x_2+\cdots\]
converges and find its sum. | "[\"e - 1\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2947, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "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:16901"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"86"
] | [
"skywork"
] | 8852afd284a6db6c18cfd38fa6866637bff6cb047ae852cbb86bf0e349132d24 | [
{
"content": "Given the fourth degree polynomial\n$$\nx^{4}-18 x^{3}+k x^{2}+200 x-1984=0\n$$\ntwo of its roots have a product of -32. Determine the value of $k$.",
"role": "user"
}
] | Given the fourth degree polynomial
$$
x^{4}-18 x^{3}+k x^{2}+200 x-1984=0
$$
two of its roots have a product of -32. Determine the value of $k$. | "[\"86\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 2955, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "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:16902"}] | {
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} |
[
"43"
] | [
"skywork"
] | ea535991711b4122df1e5cfbc9d7fa6f57bf84bc96d73c609cb282976f442271 | [
{
"content": "Find the smallest prime number that can be expressed as the sum of five different prime numbers.",
"role": "user"
}
] | Find the smallest prime number that can be expressed as the sum of five different prime numbers. | "[\"43\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2973, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "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:16903"}] | {
"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{7}{3}"
] | [
"skywork"
] | 24082de143bc1d239a561965571cb6014b97c5763df57ef2adcec936a774d549 | [
{
"content": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\in (-\\infty , m]$... | The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$. | "[\"\\\\frac{7}{3}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2940, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "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:16904"}, {"dataset": "skywork", "extra_in... | {
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"reference_components": 1,
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} |
[
"$2 \\pi + 4$"
] | [
"skywork"
] | 937686bb3e4b808c964370740787bd64e933ace3fb23fd4900bda135cf84f243 | [
{
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region?",
"role": "user"
}
] | A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region? | "[\"$2 \\\\pi + 4$\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2965, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "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:16905"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"1720"
] | [
"skywork"
] | a523408a0931671af0c151ccb4edfbe9759acea7354dadb1abf8bb01b0bc30cd | [
{
"content": "A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integer... | A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
... | "[\"1720\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2954, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16906"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
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} |
[
"-0.05"
] | [
"skywork"
] | 9cf37333183fd5b5d31230e0606fa7b44ddd2bbfa76edb137882c47d665e1dc4 | [
{
"content": "What is the largest quotient that can be formed using two numbers chosen from the set $\\{-30, -6, -1, 3, 5, 20\\}$, where one of the numbers must be negative?",
"role": "user"
}
] | What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative? | "[\"-0.05\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2963, "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:16907"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{n+1}{4}+2^{n}+(-\\frac{5}{4}n+\\frac{5}{12})3^{n}"
] | [
"skywork"
] | bf3e5fa52a24527d8290deceae635f9f21ff7ed7d0130dff01fa0df03eb7b918 | [
{
"content": "Find the general term of the sequence defined by $a_{0}=a_{1}=0, a_{n+2}=6 a_{n+1}-9 a_{n}+2^{n}+n$.",
"role": "user"
}
] | Find the general term of the sequence defined by $a_{0}=a_{1}=0, a_{n+2}=6 a_{n+1}-9 a_{n}+2^{n}+n$. | "[\"\\\\frac{n+1}{4}+2^{n}+(-\\\\frac{5}{4}n+\\\\frac{5}{12})3^{n}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2974, "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:16908"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{10}"
] | [
"skywork"
] | 40b3be43332607bfb8812adadada7e9900896a533d4ce28a68a6103dd698ed72 | [
{
"content": "14. Let $c \\in \\mathbf{C}$, the polynomial\n$$\np(x)=\\left(x^{2}-2 x+2\\right)\\left(x^{2}-c x+4\\right)\\left(x^{2}-4 x+8\\right)\n$$\n\nhas exactly four distinct complex roots. Then $|c|=(\\quad)$.\n(A) 2\n(B) $\\sqrt{6}$\n(C) $2 \\sqrt{2}$\n(D) 3\n(E) $\\sqrt{10}$",
"role": "user"
}
] | 14. Let $c \in \mathbf{C}$, the polynomial
$$
p(x)=\left(x^{2}-2 x+2\right)\left(x^{2}-c x+4\right)\left(x^{2}-4 x+8\right)
$$
has exactly four distinct complex roots. Then $|c|=(\quad)$.
(A) 2
(B) $\sqrt{6}$
(C) $2 \sqrt{2}$
(D) 3
(E) $\sqrt{10}$ | "[\"\\\\sqrt{10}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2972, "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:16909"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{3}"
] | [
"skywork"
] | c4cde71ada2f8a4f4d51c1531f5ce161b8dbcb691e846e1480f92560b173dcf5 | [
{
"content": "11. If $\\log _{4}(x+2 y)+\\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is",
"role": "user"
}
] | 11. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is | "[\"\\\\sqrt{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2952, "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:16910"}, {"dataset": "skywork", "extra_inf... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"-10 \\leqslant a \\leqslant-1"
] | [
"skywork"
] | e8a412b1af3d4a4817be2d99f7259feb8cab947a90a4fa2346f6ebdb829ecf00 | [
{
"content": "7. For all $x \\in\\left[-2, \\frac{1}{2}\\right]$, the inequality\n$$\na x^{3}-x^{2}+x+1 \\geqslant 0\n$$\n\nalways holds. Then the range of the real number $a$ is $\\qquad$ $\\because$",
"role": "user"
}
] | 7. For all $x \in\left[-2, \frac{1}{2}\right]$, the inequality
$$
a x^{3}-x^{2}+x+1 \geqslant 0
$$
always holds. Then the range of the real number $a$ is $\qquad$ $\because$ | "[\"-10 \\\\leqslant a \\\\leqslant-1\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2967, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16911"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"100"
] | [
"skywork"
] | 46eae1f6336d016630ebaeccdd1040e0e00e797f44a3a891f616e282d701d519 | [
{
"content": "1. A batch of shoes, purchased for 180 thousand rubles, was sold in the first week at a price higher than the purchase price by $25 \\%$, then the markup was reduced to $16 \\%$ of the purchase price; and the entire batch of shoes was sold for $20 \\%$ more than it was purchased for. For what amou... | 1. A batch of shoes, purchased for 180 thousand rubles, was sold in the first week at a price higher than the purchase price by $25 \%$, then the markup was reduced to $16 \%$ of the purchase price; and the entire batch of shoes was sold for $20 \%$ more than it was purchased for. For what amount was the shoes sold in ... | "[\"100\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2961, "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:16912"}] | {
"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
} |
[
"4016"
] | [
"skywork"
] | 9b317fdbbcea43dddefaa49ef22a14fcde3028905632aa73117ac66bbd4c17d7 | [
{
"content": "Berlov S.L.\n\nA convex 2011-gon is drawn on the board. Petya sequentially draws diagonals in it such that each newly drawn diagonal intersects no more than one of the previously drawn diagonals at an internal point. What is the maximum number of diagonals Petya can draw?",
"role": "user"
}
... | Berlov S.L.
A convex 2011-gon is drawn on the board. Petya sequentially draws diagonals in it such that each newly drawn diagonal intersects no more than one of the previously drawn diagonals at an internal point. What is the maximum number of diagonals Petya can draw? | "[\"4016\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2975, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16913"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{28}{3}\\pi"
] | [
"skywork"
] | 6d83d069f7326a33629a25587bd2798d1ed0528e63672a755c5dbcf25ac72200 | [
{
"content": "6. Given that the plane containing $\\triangle P A D$ is perpendicular to the plane containing rectangle $A B C D$, $P A=P D=A B=2, \\angle A P D=60^{\\circ}$, if $P, A, B, C, D$ are all on the same sphere, then the surface area of this sphere is $\\qquad$",
"role": "user"
}
] | 6. Given that the plane containing $\triangle P A D$ is perpendicular to the plane containing rectangle $A B C D$, $P A=P D=A B=2, \angle A P D=60^{\circ}$, if $P, A, B, C, D$ are all on the same sphere, then the surface area of this sphere is $\qquad$ | "[\"\\\\frac{28}{3}\\\\pi\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2959, "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:16914"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"0.20"
] | [
"skywork"
] | 944e92763b90ed198c0075046e091dde27976fcc055eeac67e17d7f642221f9d | [
{
"content": "Example 7. Calculate:\n$$\n\\frac{(742-108+321 \\times 0.082) \\times(37.2-28.7)}{517.4 \\times(90.28-37.2)}\n$$",
"role": "user"
}
] | Example 7. Calculate:
$$
\frac{(742-108+321 \times 0.082) \times(37.2-28.7)}{517.4 \times(90.28-37.2)}
$$ | "[\"0.20\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2978, "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:16915"}] | {
"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"
] | 99030dffe4de5e7b36cbcb86baf6091b05ad65c9edf268fa853d11d7638bfc09 | [
{
"content": "4. Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \\cdots, A_{n}$ satisfying $A_{1} \\cup A_{2} \\cup \\cdots \\cup A_{n}=\\mathbf{N}^{*}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-c \\geqslan... | 4. Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfying $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=\mathbf{N}^{*}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-c \geqslant a^{i}$. | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2970, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16916"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"25"
] | [
"skywork"
] | 2783c3daa8f9d9f4f7f9dd948486b14b464d61e8fb2eed2b44cb5ea26843d4a4 | [
{
"content": "What is the value of $3.5 \\times 2.5+6.5 \\times 2.5$ ?",
"role": "user"
}
] | What is the value of $3.5 \times 2.5+6.5 \times 2.5$ ? | "[\"25\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2986, "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:16917"}] | {
"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{13-2 \\sqrt{30}}{7}"
] | [
"skywork"
] | 27e2649dd8c2f5a77044c06ae2df2535eca72798ab3f244f37298f897dd6a539 | [
{
"content": "3. The solution to the equation $\\sqrt{x^{2}-\\frac{1}{2} x+1}+\\sqrt{x^{2}-\\frac{2}{3} x+1}$ $=1+\\frac{\\sqrt{30}}{6}$ is . $\\qquad$",
"role": "user"
}
] | 3. The solution to the equation $\sqrt{x^{2}-\frac{1}{2} x+1}+\sqrt{x^{2}-\frac{2}{3} x+1}$ $=1+\frac{\sqrt{30}}{6}$ is . $\qquad$ | "[\"\\\\frac{13-2 \\\\sqrt{30}}{7}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2971, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16918"}] | {
"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
} |
[
"7744"
] | [
"skywork"
] | 20964aa96597b7465360dea8fee31c8eccddd22dfb91564b38342d52a4373a8a | [
{
"content": "A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers.",
"role": "user"
}
] | A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. | "[\" 7744 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2962, "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:16919"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{6}"
] | [
"skywork"
] | 9526e4f4580bb7926be51484e6a08bc5a27d378d7f3733bbb4b01ddf8cdf8c9f | [
{
"content": "II. (40 points) Given positive real numbers $x, y, z$ satisfying $x+y+z=1$. Try to find the minimum value of the real number $k$ such that the inequality\n$$\n\\frac{x^{2} y^{2}}{1-z}+\\frac{y^{2} z^{2}}{1-x}+\\frac{z^{2} x^{2}}{1-y} \\leqslant k-3 x y z\n$$\n\nalways holds.",
"role": "user"
... | II. (40 points) Given positive real numbers $x, y, z$ satisfying $x+y+z=1$. Try to find the minimum value of the real number $k$ such that the inequality
$$
\frac{x^{2} y^{2}}{1-z}+\frac{y^{2} z^{2}}{1-x}+\frac{z^{2} x^{2}}{1-y} \leqslant k-3 x y z
$$
always holds. | "[\"\\\\frac{1}{6}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2969, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16920"}] | {
"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
} |
[
"50"
] | [
"skywork"
] | 0262a3fb424466e3d18319dd8aff729d4ae91609bb0227df3bc01692d7d61ea0 | [
{
"content": "If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$?",
"role": "user"
}
] | If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$? | "[\"50\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2988, "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:16921"}] | {
"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
} |
[
"400"
] | [
"skywork"
] | fcda3e414c1d53d12308041ac6d2870fa3fb4e035f6b44ef6c11709cabbf5b6e | [
{
"content": "19.2.1 * Find the sum of all positive rational numbers less than 10 that, when written as reduced fractions, have a denominator of 30.",
"role": "user"
}
] | 19.2.1 * Find the sum of all positive rational numbers less than 10 that, when written as reduced fractions, have a denominator of 30. | "[\"400\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2982, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16922"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"5.0833"
] | [
"skywork"
] | b8b4e2d3612b9fa2454ec2dd74ec9be3369bb00529b25cb65789a13cb09e2b6e | [
{
"content": "A graph shows the number of books read in June by the top readers in a school library. The data points given are:\n- 4 readers read 3 books each\n- 5 readers read 5 books each\n- 2 readers read 7 books each\n- 1 reader read 10 books\nDetermine the mean (average) number of books read by these reade... | A graph shows the number of books read in June by the top readers in a school library. The data points given are:
- 4 readers read 3 books each
- 5 readers read 5 books each
- 2 readers read 7 books each
- 1 reader read 10 books
Determine the mean (average) number of books read by these readers. | "[\"5.0833\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2984, "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:16923"}] | {
"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"
] | 372b19ba7d4cc902f6f3d915358a4691510b93fb6b5aaf3d7aad2be8da3d688b | [
{
"content": "If $y=77$, then $\\frac{7 y+77}{77}$ is equal to\n(A) 8\n(B) 12\n(C) 78\n(D) 84\n(E) 540",
"role": "user"
}
] | If $y=77$, then $\frac{7 y+77}{77}$ is equal to
(A) 8
(B) 12
(C) 78
(D) 84
(E) 540 | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2985, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "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:16924"}] | {
"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
} |
[
"5044"
] | [
"skywork"
] | 4c3fbd2f2998401ea564ad57fc70b45997b0d7582c0a070108c6497af1b7ef80 | [
{
"content": "5. The sequence $\\left\\{a_{n}\\right\\}$ satisfies: $a_{1}=\\frac{1}{4}, a_{2}=\\frac{1}{5}$ and $a_{1} a_{2}+a_{2} a_{3}+\\cdots+a_{n} a_{n+1}=n a_{1} a_{n+1}$ for any positive integer $n$, then $\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{97}}$ is $\\qquad$ .",
"role": "user"
... | 5. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{4}, a_{2}=\frac{1}{5}$ and $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n} a_{n+1}=n a_{1} a_{n+1}$ for any positive integer $n$, then $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{97}}$ is $\qquad$ . | "[\"5044\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2948, "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:16925"}] | {
"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"
] | e0c17f4f3eed73cd544600da117094d1cc0ece35f02eb302bdb1b0c895f0857b | [
{
"content": "If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$?",
"role": "user"
}
] | If $f(x)=ax+b$ and $f^{-1}(x)=bx+a$ with $a$ and $b$ real, what is the value of $a+b$? | "[\"-2\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2993, "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:16926"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | 0aa8fdaa63813eefdaa01a6ca9d3e6c90d6af7ae624cb1c2c1966b24fbc28eb8 | [
{
"content": "294. $\\int_{1}^{e} \\frac{3 \\ln ^{2} x d x}{x}$.\n\nTranslate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n294. $\\int_{1}^{e} \\frac{3 \\ln ^{2} x \\, dx}{x}$.",
"role": "user"
}
] | 294. $\int_{1}^{e} \frac{3 \ln ^{2} x d x}{x}$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
294. $\int_{1}^{e} \frac{3 \ln ^{2} x \, dx}{x}$. | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 3011, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16927"}] | {
"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^{1/8}"
] | [
"skywork"
] | bc4e617c4523a94f0ac1ee6c8443fc28b72994bbaf857bcdae812b9fcd390073 | [
{
"content": "Let $ (x_1,x_2,\\cdots)$ be a sequence of positive numbers such that $ (8x_2 \\minus{} 7x_1)x_1^7 \\equal{} 8$ and\r\n\\[ x_{k \\plus{} 1}x_{k \\minus{} 1} \\minus{} x_k^2 \\equal{} \\frac {x_{k \\minus{} 1}^8 \\minus{} x_k^8}{x_k^7x_{k \\minus{} 1}^7} \\text{ for }k \\equal{} 2,3,\\ldots\r\n\... | Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such th... | "[\"8^{1/8}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2980, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "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:16928"}, {"dataset": "skywork", "extra_inf... | {
"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
} |
[
"a = 4"
] | [
"skywork"
] | 9fe36f7a21010e8ef640428dd98b6f0178bc883a37e866ff16331288ab0e1b6b | [
{
"content": "Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \\frac{3}{2}x + \\frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original proposition is true, while... | Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \frac{3}{2}x + \frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original proposition is true, while its converse is false, ... | "[\"a = 4\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2950, "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:16929"}] | {
"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
} |
[
"75"
] | [
"skywork"
] | e4e236e7157ccd36b26c12856fa9ba469788e7de4cde50df10afe48b06276174 | [
{
"content": "What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$?",
"role": "user"
}
] | What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$? | "[\"75\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 3012, "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:16930"}] | {
"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
} |
[
"12"
] | [
"skywork"
] | 9dd8c9c69e468c8b9803aa59dea8bde45541eacfb3ee058561d09703b27dab54 | [
{
"content": "In a candy dish, the ratio of red to blue candies is $3: 5$. When 3 blue candies are removed, the ratio of red to blue candies becomes $2: 3$. How many more blue candies than red candies were in the dish before any candies were removed?\n(A) 10\n(B) 12\n(C) 6\n(D) 8\n(E) 14",
"role": "user"
... | In a candy dish, the ratio of red to blue candies is $3: 5$. When 3 blue candies are removed, the ratio of red to blue candies becomes $2: 3$. How many more blue candies than red candies were in the dish before any candies were removed?
(A) 10
(B) 12
(C) 6
(D) 8
(E) 14 | "[\"12\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2987, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "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:16931"}] | {
"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
} |
[
"3"
] | [
"skywork"
] | 96dc6482cb2f1cc3c6571aa888fc4267724bc76286f980387251a0a88e4d1653 | [
{
"content": "Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7.",
"role": "user"
}
] | Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. | "[\"3\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2995, "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:16932"}] | {
"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
} |
[
"41"
] | [
"skywork"
] | a0dc9fd6916a51e5ebf612c21109ffa2a2f1015129ec60117e352ee128cd41cc | [
{
"content": "If $x$ is real, compute the maximum integer value of\n\\[\\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\\]",
"role": "user"
}
] | If $x$ is real, compute the maximum integer value of
\[\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\] | "[\"41\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 3003, "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:16933"}] | {
"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 {5}{2}"
] | [
"skywork"
] | 4b8f486ce2288c96fd8135bc1ded7d7247de6262ef470083f66ea00afef3cb43 | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c\\sin A= \\sqrt {3}a\\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\\sin x\\cos ( \\frac {π}{2}-x)- \\sqrt {3}\\sin (π+x)\\cos x+\\sin ( \\frac {π}{2}+x)\\cos x$.\n... | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c\sin A= \sqrt {3}a\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\sin x\cos ( \frac {π}{2}-x)- \sqrt {3}\sin (π+x)\cos x+\sin ( \frac {π}{2}+x)\cos x$.
(1) Find the period and the equation... | "[\"\\\\frac {5}{2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2964, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:16934"}] | {
"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
} |
[
"141"
] | [
"skywork"
] | 8083249bcd409653664c1d35920d0fcdfd055c8781cdef6c5b7ebf03b82c1b55 | [
{
"content": "Let $AD, BE$, and $CF$ be segments sharing a common midpoint, with $AB < AE$ and $BC < BF$. Suppose that each pair of segments forms a $60^{\\circ}$ angle, and that $AD=7, BE=10$, and $CF=18$. Let $K$ denote the sum of the areas of the six triangles $\\triangle ABC, \\triangle BCD, \\triangle CDE,... | Let $AD, BE$, and $CF$ be segments sharing a common midpoint, with $AB < AE$ and $BC < BF$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $AD=7, BE=10$, and $CF=18$. Let $K$ denote the sum of the areas of the six triangles $\triangle ABC, \triangle BCD, \triangle CDE, \triangle DEF, \triangle ... | "[\"141\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2981, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16935"}] | {
"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
} |
[
"120"
] | [
"skywork"
] | 217807896958ccebccc6fa95e5e4e0160c26267f805d0a2fb6b1ca636cec3723 | [
{
"content": "Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$",
"role": "user"
}
] | Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$ | "[\"120\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2992, "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:16936"}] | {
"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
} |
[
"41"
] | [
"skywork"
] | bd17c5ed702211aae134134f13ef46baa1fd15da07e6963809e483db2f777db3 | [
{
"content": "1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\\qquad$ apples in the first box.",
"role": "user"
}
] | 1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\qquad$ apples in the first box. | "[\"41\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2989, "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:16937"}] | {
"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
} |
[
"183"
] | [
"skywork"
] | dd5294871e94a8e8279ce281a68061f126a1f1a0db51d1a4662b491c106209be | [
{
"content": "A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in... | A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of th... | "[\"183\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 2976, "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:16938"}, {"dataset": "skywork", "extra_... | {
"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
} |
[
"1771"
] | [
"skywork"
] | c27dee9fe32d9bf34eb45abe251bfd84334f1e35cff94ab5f49ad503be6e0ccf | [
{
"content": "Compute\n\\[\\prod_{n = 1}^{20} \\frac{n + 3}{n}.\\]",
"role": "user"
}
] | Compute
\[\prod_{n = 1}^{20} \frac{n + 3}{n}.\] | "[\"1771\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 3004, "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:16939"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"{\\frac{1}{4},\\frac{3}{2},\\frac{2}{3},4}"
] | [
"skywork"
] | 727ca644ecdf5655952665cdb75bfa58f715f91246111682057c52d01c7ad924 | [
{
"content": "(5) If the set $A=\\left\\{z \\left\\lvert\\, z=\\frac{p}{q}\\right., p+q=5, p, q \\in \\mathbf{N}^{*}\\right\\}$, then the set $A=$ $\\qquad$ .",
"role": "user"
}
] | (5) If the set $A=\left\{z \left\lvert\, z=\frac{p}{q}\right., p+q=5, p, q \in \mathbf{N}^{*}\right\}$, then the set $A=$ $\qquad$ . | "[\"{\\\\frac{1}{4},\\\\frac{3}{2},\\\\frac{2}{3},4}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2990, "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:16940"}] | {
"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
} |
[
"34"
] | [
"skywork"
] | 93bf66c7cd697af967df15d19cc919230978f5edb26fd3df669af7dc5ad97336 | [
{
"content": "For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$?",
"role": "user"
}
] | For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$? | "[\"34\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2998, "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:16941"}] | {
"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
} |
[
"100"
] | [
"skywork"
] | 258d143161b9b46e0ea0e06ed6d3fd0285fc6441a77857ae2cfdd099fe40f1fa | [
{
"content": "1. The quadratic trinomial $a x^{2}+b x+c$, for $x=1 ; 2 ; 3$, takes values $0 ; 1 ; 4$ respectively. What value does this polynomial take for $x=11$?",
"role": "user"
}
] | 1. The quadratic trinomial $a x^{2}+b x+c$, for $x=1 ; 2 ; 3$, takes values $0 ; 1 ; 4$ respectively. What value does this polynomial take for $x=11$? | "[\"100\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2997, "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:16942"}] | {
"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
} |
[
"\\alpha=8"
] | [
"skywork"
] | affabb467916907cfe414ae983013fc6e940e9724c67ded4ee8b2c110dcd3d31 | [
{
"content": "A4\n\nLet $M=\\left\\{(a, b, c) \\in \\mathbb{R}^{3}: 0<a, b, c<\\frac{1}{2}\\right.$ with $\\left.a+b+c=1\\right\\}$ and $f: M \\rightarrow \\mathbb{R}$ given as\n\n$$\nf(a, b, c)=4\\left(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\right)-\\frac{1}{a b c}\n$$\n\nFind the best (real) bounds $\\alpha$ ... | A4
Let $M=\left\{(a, b, c) \in \mathbb{R}^{3}: 0<a, b, c<\frac{1}{2}\right.$ with $\left.a+b+c=1\right\}$ and $f: M \rightarrow \mathbb{R}$ given as
$$
f(a, b, c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{a b c}
$$
Find the best (real) bounds $\alpha$ and $\beta$ such that
$$
f(M)=\{f(a, b, c):(a, ... | "[\"\\\\alpha=8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2979, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "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:16944"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"skywork"
] | a48ac0200b486a7454e3ad67bc92272850f333b1de3f497f19cbc86aa54d9128 | [
{
"content": "If $m$ and $n$ are integers greater than zero and $m < n$, we define $m \\nabla n$ as the sum of the integers between $m$ and $n$, including $m$ and $n$. For example, $5 \\nabla 8 = 5 + 6 + 7 + 8 = 26$. What is the value of $\\frac{22 \\nabla 26}{4 \\nabla 6}$?\n(a) 4\n(b) 6\n(c) 8\n(d) 10\n(e) 12... | If $m$ and $n$ are integers greater than zero and $m < n$, we define $m \nabla n$ as the sum of the integers between $m$ and $n$, including $m$ and $n$. For example, $5 \nabla 8 = 5 + 6 + 7 + 8 = 26$. What is the value of $\frac{22 \nabla 26}{4 \nabla 6}$?
(a) 4
(b) 6
(c) 8
(d) 10
(e) 12 | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2991, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16945"}] | {
"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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