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 |
|---|---|---|---|---|---|---|---|---|
[
"-\\frac{7\\sqrt{2}}{10}"
] | [
"skywork"
] | 50047822d2cf98ee7707b4553b16e1e13d864abf6aea1691c523e85e5284be14 | [
{
"content": "Given $f(x) = 2\\sin x (\\sin x + \\cos x)$, where $x \\in \\mathbb{R}$.\n\n(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.\n\n(Ⅱ) If $f\\left( \\frac{a}{2} \\right) = 1 + \\frac{3\\sqrt{2}}{5}$, and $\\frac{3\\pi}{4} < a < \\frac{5\\pi}{4}$, find the value of $\\cos a$.",
... | Given $f(x) = 2\sin x (\sin x + \cos x)$, where $x \in \mathbb{R}$.
(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.
(Ⅱ) If $f\left( \frac{a}{2} \right) = 1 + \frac{3\sqrt{2}}{5}$, and $\frac{3\pi}{4} < a < \frac{5\pi}{4}$, find the value of $\cos a$. | "[\"-\\\\frac{7\\\\sqrt{2}}{10}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 109198, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "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:119099"}] | {
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[
"0or-3"
] | [
"skywork"
] | 59f7a84dd75788a240b3e01168dcc3eb94f28102e1fab0ce066c382dcddb0995 | [
{
"content": "4.1. Given is the polynomial $P(x)=x^{2}-\\alpha$, where $\\alpha$ is a real number. Determine all polynomials of the form $Q(x)=x^{3}+a x^{2}+b x+c$, where $a, b$, and $c$ are real numbers, such that for every real number $x$, $P(Q(x))=Q(P(x))$ holds.",
"role": "user"
}
] | 4.1. Given is the polynomial $P(x)=x^{2}-\alpha$, where $\alpha$ is a real number. Determine all polynomials of the form $Q(x)=x^{3}+a x^{2}+b x+c$, where $a, b$, and $c$ are real numbers, such that for every real number $x$, $P(Q(x))=Q(P(x))$ holds. | "[\"0or-3\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 109190, "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:119101"}] | {
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[
"369 \\sqrt{3} - 540"
] | [
"skywork"
] | 4a7ec3fdbe48c14e6010b44412b05e840f2aeccac4fa98306b7ec09a9abd8640 | [
{
"content": "The sides of rectangle $ABCD$ have lengths $12$ (height) and $15$ (width). An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle.",
"role": "user"
}
] | The sides of rectangle $ABCD$ have lengths $12$ (height) and $15$ (width). An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. | "[\"369 \\\\sqrt{3} - 540\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 109199, "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:119102"}] | {
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} |
[
"\\frac{39}{5}"
] | [
"skywork"
] | 1cad994313b88f1b99b4f01bebc3d0bda0f890497b2d0ca2b4e2f70844482d1a | [
{
"content": "## Task B-1.7.\n\nTwo right triangles $P S T$ and $R S T$ have a common hypotenuse $\\overline{S T}$. Vertices $P$ and $R$ are on the same side of the hypotenuse $\\overline{S T}$. The legs $\\overline{P T}$ and $\\overline{R S}$ intersect at point $Q$. If $|P S|=6 \\mathrm{~cm},|P T|=17 \\mathrm{... | ## Task B-1.7.
Two right triangles $P S T$ and $R S T$ have a common hypotenuse $\overline{S T}$. Vertices $P$ and $R$ are on the same side of the hypotenuse $\overline{S T}$. The legs $\overline{P T}$ and $\overline{R S}$ intersect at point $Q$. If $|P S|=6 \mathrm{~cm},|P T|=17 \mathrm{~cm}$ and $|R T|=1 \mathrm{~cm... | "[\"\\\\frac{39}{5}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 109206, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:119103"}] | {
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"reference_present": true
} |
[
"2520"
] | [
"skywork"
] | f4459bd8e912ee579d95b2c6c019297d9a9c717f675ee691f303e83d4ae5d673 | [
{
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\\{b_{n}\\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>$(2)$ ... | Given an arithmetic sequence $\{a_{n}\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\{b_{n}\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let the sum of the first $n$ ... | "[\"2520\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 109194, "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:119105"}] | {
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} |
[
"12"
] | [
"skywork"
] | 6c6a3713034a55feb842780a0ead474f7d0f37642b3c57598e9f738eafee1af3 | [
{
"content": "5. For any positive integer $n$, let $a_{n}$ denote the number of triangles with integer side lengths and the longest side length being $2 n$.\n(1) Find the expression for $a_{n}$ in terms of $n$;\n(2) Let the sequence $\\left\\{b_{n}\\right\\}$ satisfy\n$$\n\\sum_{k=1}^{n}(-1)^{n-k} \\mathrm{C}_{... | 5. For any positive integer $n$, let $a_{n}$ denote the number of triangles with integer side lengths and the longest side length being $2 n$.
(1) Find the expression for $a_{n}$ in terms of $n$;
(2) Let the sequence $\left\{b_{n}\right\}$ satisfy
$$
\sum_{k=1}^{n}(-1)^{n-k} \mathrm{C}_{n}^{k} b_{k}=a_{n}\left(n \in \m... | "[\"12\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 109197, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:119106"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"240"
] | [
"skywork"
] | 64adeac53a3f33a70685cffcb6f9b3add69e8b8d0ea20cb998e497a9a3b32f65 | [
{
"content": "The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that th... | The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that the seventh digit is 9 m... | "[\"240\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 109202, "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:119108"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3+ \\sqrt{3}}{4}"
] | [
"skywork"
] | 1e54911d7ed82ee60bd727dccbc644ad08350000c53c773e8fc695c2d8b7ab0e | [
{
"content": "Given that the three interior angles $A$, $B$, $C$ of $\\triangle ABC$ form an arithmetic sequence, and the side $b$ opposite to angle $B$ equals $\\sqrt{3}$, and the function $f(x)=2 \\sqrt{3}\\sin ^{2}x+2\\sin x\\cos x- \\sqrt{3}$ reaches its maximum value at $x=A$, then the area of $\\triangle ... | Given that the three interior angles $A$, $B$, $C$ of $\triangle ABC$ form an arithmetic sequence, and the side $b$ opposite to angle $B$ equals $\sqrt{3}$, and the function $f(x)=2 \sqrt{3}\sin ^{2}x+2\sin x\cos x- \sqrt{3}$ reaches its maximum value at $x=A$, then the area of $\triangle ABC$ is __________. | "[\"\\\\frac{3+ \\\\sqrt{3}}{4}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 109191, "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:119109"}] | {
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} |
[
"\\text{No}"
] | [
"skywork"
] | be8a6fd1a0b52e85728cb2f4d91a07e48dae7290e6919bc1e73efe2ac215f74f | [
{
"content": "De\ffine the [i]quasi-primes[/i] as follows.\n$\\bullet$ The \ffirst quasi-prime is $q_1 = 2$\n$\\bullet$ For $n \\ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \\le i \\le j \\le n - 1$.\nDetermine, with proof, whethe... | De fine the [i]quasi-primes[/i] as follows.
$\bullet$ The first quasi-prime is $q_1 = 2$
$\bullet$ For $n \ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$.
Determine, with proof, whether or not $1000$ is a quasi-prime. | "[\"\\\\text{No}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 109201, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:119111"}] | {
"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
} |
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