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{11}{3}"
] | [
"skywork"
] | 0375634157290c067971bc5ba8050ae6d45a326d46d94470af8685ee222c3c2d | [
{
"content": "2. Acute $\\triangle A B C$ is inscribed in $\\odot O$ with radius $R$, $H$ is the orthocenter of $\\triangle A B C$, and the extension of $A O$ intersects $B C$ at point $M$. If $O H \\perp A O, B C=10, O A=6$, then $O M=$ $\\qquad$ -",
"role": "user"
}
] | 2. Acute $\triangle A B C$ is inscribed in $\odot O$ with radius $R$, $H$ is the orthocenter of $\triangle A B C$, and the extension of $A O$ intersects $B C$ at point $M$. If $O H \perp A O, B C=10, O A=6$, then $O M=$ $\qquad$ - | "[\"\\\\frac{11}{3}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 107933, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117887"}] | {
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[
"\\frac{4}{63}"
] | [
"skywork"
] | 8d390886779c46b003d3ddfb0ecaf2f080a2fcc9ad88e960765e1e7ec1a8c59a | [
{
"content": "13. There are two bags, $A$ and $B$. Bag $A$ contains 6 cards, one of which is labeled 0, two are labeled 1, and three are labeled 2; Bag $B$ contains 7 cards, four of which are labeled 0, one is labeled 1, and two are labeled 2. One card is drawn from bag $A$ and two cards are drawn from bag $B$,... | 13. There are two bags, $A$ and $B$. Bag $A$ contains 6 cards, one of which is labeled 0, two are labeled 1, and three are labeled 2; Bag $B$ contains 7 cards, four of which are labeled 0, one is labeled 1, and two are labeled 2. One card is drawn from bag $A$ and two cards are drawn from bag $B$, making a total of thr... | "[\"\\\\frac{4}{63}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107942, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117888"}] | {
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[
"13205"
] | [
"skywork"
] | d8a1e0b2bd5e748286594428bc352bfff04e9961e68ea4efbc53ac9c347387f3 | [
{
"content": "3. Arrange all positive integers whose digits do not exceed 5 in ascending order. What is the 2021st number?",
"role": "user"
}
] | 3. Arrange all positive integers whose digits do not exceed 5 in ascending order. What is the 2021st number? | "[\"13205\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107953, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117889"}] | {
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[
"182"
] | [
"skywork"
] | 9f4da8e126d95a10800e3faed8bc2aec0915b3cfb22639819f48d363978d954e | [
{
"content": "By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called \"nice\" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?",
"role": "user"... | By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | "[\"182\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 107959, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117890"}] | {
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[
"16"
] | [
"skywork"
] | d68db24764300aea9424f7abdf2f81411b399c896c9174c2d9b15e7a2074e2d3 | [
{
"content": "In a group of $n$ people, each one had a different ball. They performed a sequence of swaps; in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible numb... | In a group of $n$ people, each one had a different ball. They performed a sequence of swaps; in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: $a$ )... | "[\"16\"]" | train-math-numinamath1.5_olympiads_ref | [{"dataset": "skywork", "extra_info": {"index": 107943, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 10, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117891"}] | {
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} |
[
"40"
] | [
"skywork"
] | c614e5edf2a9e78329554b13b54201d6a4455776ef617b75571cd7914346cd63 | [
{
"content": "A circle has an area of $M\\text{ cm}^2$ and a circumference of $N\\text{ cm}$. If $\\dfrac{M}{N}=20$, what is the radius of the circle, in cm?",
"role": "user"
}
] | A circle has an area of $M\text{ cm}^2$ and a circumference of $N\text{ cm}$. If $\dfrac{M}{N}=20$, what is the radius of the circle, in cm? | "[\"40\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107950, "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:117893"}] | {
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} |
[
"760"
] | [
"skywork"
] | abf6f38151196fa47a2cdb227a9a242579084ad1943dbfc854ab5e69c8f3203d | [
{
"content": "Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \\cap B| \\equal{} |B \\cap C| \\equal{} |C \\cap A| \\equal{} 1$ and $ A \\cap B \\cap C \\equal{} \\emptyset$. For example, $ (\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $ N$ be the ... | Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$. For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting order... | "[\"760\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107947, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "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:117894"}] | {
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} |
[
"30"
] | [
"skywork"
] | 1f0ed4a6c8a8264bed0847037c80cd5ca78da899bed2f2507c42665c83b89766 | [
{
"content": "In the following diagram, \\(ABCD\\) is a square, \\(BD \\parallel CE\\) and \\(BE = BD\\). Let \\(\\angle E = x^{\\circ}\\). Find \\(x\\).",
"role": "user"
}
] | In the following diagram, \(ABCD\) is a square, \(BD \parallel CE\) and \(BE = BD\). Let \(\angle E = x^{\circ}\). Find \(x\). | "[\"30\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107944, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117895"}] | {
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} |
[
"-15"
] | [
"skywork"
] | 5b99aeea33073143776494c70b89ff706ad1e03500b8c69b5e783cc88ffa024c | [
{
"content": "The maximum and minimum values of the function $y=2x^3-3x^2-12x+5$ in the interval $[0,3]$ are respectively what are the values?",
"role": "user"
}
] | The maximum and minimum values of the function $y=2x^3-3x^2-12x+5$ in the interval $[0,3]$ are respectively what are the values? | "[\"-15\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107982, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117897"}] | {
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} |
[
"\\frac{n-1 + \\sqrt{n-1}}{\\sqrt{n}}"
] | [
"skywork"
] | 00443e4210d84aa7b8a8f71f64cd457170e0d0288d21f9d0630c47f8fab21863 | [
{
"content": "Given integer $n\\geq 2$. Find the minimum value of $\\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\\cdots$, ${a_n}$ and ${b}$,\n$$\\lambda\\sum\\limits_{i=1}^n\\sqrt{|a_i-b|}+\\sqrt{n\\left|\\sum\\limits_{i=1}^na_i\\right|}\\geqslant\\sum\\limits_{i=1}^n\\sqrt{|a_i|}.$$",
"ro... | Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$,
$$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$ | "[\"\\\\frac{n-1 + \\\\sqrt{n-1}}{\\\\sqrt{n}}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107956, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "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:117898"}] | {
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} |
[
"23"
] | [
"skywork"
] | 08c339eaad0e0a5b9c7033f476f158f3a8701d19e8c9c3ef8a4b21ddcafa34a2 | [
{
"content": "How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$?",
"role": "user"
}
] | How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | "[\"23\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107978, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "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:117899"}] | {
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} |
[
"48"
] | [
"skywork"
] | 2a579ebc70b5be3256f222117004df687f93485a8ce43858284ef7595838b614 | [
{
"content": "Four. (50 points) Let $S \\subseteq\\{1,2,3, \\cdots, 100\\}$. It is known that for any two distinct elements $a, b$ in $S$, there exist a positive integer $k$ and two distinct elements $c, d$ in $S$ (which may equal $a$ or $b$), such that $c<d$, and $a+b=c^{k} d$. Find the maximum number of eleme... | Four. (50 points) Let $S \subseteq\{1,2,3, \cdots, 100\}$. It is known that for any two distinct elements $a, b$ in $S$, there exist a positive integer $k$ and two distinct elements $c, d$ in $S$ (which may equal $a$ or $b$), such that $c<d$, and $a+b=c^{k} d$. Find the maximum number of elements in the set $S$.
保留源文本... | "[\"48\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107968, "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:117901"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"3.75"
] | [
"skywork"
] | e1700a3f3ba04bdb67abd26df72f8a48eb20cc0a34f52882048c14e0b35f7f87 | [
{
"content": "Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have.",
"role": "user"
}
] | Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have. | "[\"3.75\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107961, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "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:117902"}] | {
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} |
[
"171"
] | [
"skywork"
] | 855a171f8b77c022fd3bee4b59121405a19a784eddd28810c9d33c820fefa740 | [
{
"content": "Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score?",
"role": "user"
}
] | Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score? | "[\"171\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107985, "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:117903"}] | {
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"reference_components": 1,
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} |
[
"(\\sqrt{2}-1):1"
] | [
"skywork"
] | f962f252ad9f965c28663fc878c9854c93a3805bc559b331e908002ba196d2c1 | [
{
"content": "1.19. Find the ratio of the radius of the circle inscribed in an isosceles right triangle to the height dropped to the hypotenuse.",
"role": "user"
}
] | 1.19. Find the ratio of the radius of the circle inscribed in an isosceles right triangle to the height dropped to the hypotenuse. | "[\"(\\\\sqrt{2}-1):1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107966, "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:117904"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | cf55125d37860189130ef442f447fb192dac521c2a203442ee8ed2749b61c8ef | [
{
"content": "6. On the edges of a trihedral angle with vertex at point $S$, there are points $M, N$, and $K$ such that $S M^{2}+S N^{2}+S K^{2} \\leq 12$. Find the area of triangle $S M N$, given that the angle $M S N$ is $30^{\\circ}$, and the volume of the pyramid $S M N K$ is maximally possible.",
"role... | 6. On the edges of a trihedral angle with vertex at point $S$, there are points $M, N$, and $K$ such that $S M^{2}+S N^{2}+S K^{2} \leq 12$. Find the area of triangle $S M N$, given that the angle $M S N$ is $30^{\circ}$, and the volume of the pyramid $S M N K$ is maximally possible. | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107938, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "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:117905"}] | {
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} |
[
"96.5"
] | [
"skywork"
] | 2b57b0d0879f0d4c86fe911a9b55ad74dac3ae2df9d772e55d94412d52c3ba17 | [
{
"content": "15. Let $I$ be the in-centre of $\\triangle A B C$. If $B C=A C+A I$ and $\\angle A B C-\\angle A C B=13^{\\circ}$, find $\\angle B A C$.\n(2 marks)\n設 $I$ 為 $\\triangle A B C$ 的內心。若 $B C=A C+A I$, 且 $\\angle A B C-\\angle A C B=13^{\\circ}$, 求 $\\angle B A C$.",
"role": "user"
}
] | 15. Let $I$ be the in-centre of $\triangle A B C$. If $B C=A C+A I$ and $\angle A B C-\angle A C B=13^{\circ}$, find $\angle B A C$.
(2 marks)
設 $I$ 為 $\triangle A B C$ 的內心。若 $B C=A C+A I$, 且 $\angle A B C-\angle A C B=13^{\circ}$, 求 $\angle B A C$. | "[\"96.5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107960, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117906"}] | {
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} |
[
"(t, t, 4t), (2t, t, 6t), (t, 2t, 6t)"
] | [
"skywork"
] | b850f5c816271524b2c6ddca1db42d1fec8b206ce25ca0f5ccdb494ac791a14d | [
{
"content": "Find all triples $(a, b, n)$ of positive integers such that $a$ and $b$ are both divisors of $n$, and $a+b = \\frac{n}{2}$ .",
"role": "user"
}
] | Find all triples $(a, b, n)$ of positive integers such that $a$ and $b$ are both divisors of $n$, and $a+b = \frac{n}{2}$ . | "[\" (t, t, 4t), (2t, t, 6t), (t, 2t, 6t) \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107965, "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:117907"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
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} |
[
"96"
] | [
"skywork"
] | dd6b428ad25cf97851cc2fc15e8315c30e611d65d25ce188176a77e2b18f9acf | [
{
"content": "When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder.",
"role": "user"
}
] | When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder. | "[\"96\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107976, "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:117908"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"280"
] | [
"skywork"
] | 855d939e7a1ae52e21020f05129a904ba4110ad602d852381e7ac7ee6e9b69f1 | [
{
"content": "Let $A B C D$ be a parallelogram with $A B=480, A D=200$, and $B D=625$. The angle bisector of $\\angle B A D$ meets side $C D$ at point $E$. Find $C E$.",
"role": "user"
}
] | Let $A B C D$ be a parallelogram with $A B=480, A D=200$, and $B D=625$. The angle bisector of $\angle B A D$ meets side $C D$ at point $E$. Find $C E$. | "[\"280\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107969, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117909"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{2}"
] | [
"skywork"
] | 51f098f965b726e7253a8e27f2f337e68be25f3bdb86106cbe180879c20213b0 | [
{
"content": "4. Calculate $\\sqrt{4+\\sqrt{7}}-\\sqrt{4-\\sqrt{7}}$.",
"role": "user"
}
] | 4. Calculate $\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}$. | "[\"\\\\sqrt{2}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107992, "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:117910"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2\\frac{1}{2}"
] | [
"skywork"
] | 5fc62e8f4d2b9b89be7c7963588782617220bc432c7696e0274446e62a0bc316 | [
{
"content": "The average of four different positive whole numbers is $4.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?",
"role": "user"
}
] | The average of four different positive whole numbers is $4.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers? | "[\"2\\\\frac{1}{2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107975, "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:117912"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"70"
] | [
"skywork"
] | 87fc630e1f1dc0d98cec30b235b74b42b1ff5eea67600bdae19424a02c13b9e1 | [
{
"content": "Problem 1. Find the smallest positive integer $n$ with the following property : There does not exist an arithmetic progression of 1999 terms of real numbers containing exactly $n$ integers.",
"role": "user"
}
] | Problem 1. Find the smallest positive integer $n$ with the following property : There does not exist an arithmetic progression of 1999 terms of real numbers containing exactly $n$ integers. | "[\"70\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107972, "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:117913"}] | {
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} |
[
"1997=3\\cdot333+3\\cdot333-3:3"
] | [
"skywork"
] | 12db869b3c31bc520b71101cee7d908deaa581e0e5d6fdb4cf50adc26632a1bd | [
{
"content": "1. Write the number 1997 using 10 threes and arithmetic operations.",
"role": "user"
}
] | 1. Write the number 1997 using 10 threes and arithmetic operations. | "[\"1997=3\\\\cdot333+3\\\\cdot333-3:3\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107988, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117914"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
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} |
[
"44.5"
] | [
"skywork"
] | 2c59bde77bd2fa711d7d58ed2828d95fa2ba639e0aa6e92a6426777d5dfd7cab | [
{
"content": "Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$.",
"role": "user"
}
] | Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$. | "[\"44.5\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108003, "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:117915"}] | {
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"reference_components": 1,
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} |
[
"12500"
] | [
"skywork"
] | 2a0acd9f66550c5f2472c568134d700c6f2346bd7fcbf3b7631e840e99e3f450 | [
{
"content": "Explore the pattern, observe the following equations, and answer the questions.\n$1=1^{2}$;\n$1+3=4=2^{2}$;\n$1+3+5=9=3^{2}$;\n$1+3+5+7=16=4^{2}$;\n$1+3+5+7+9=25=5^{2}$;\n$\\ldots $\n$(1)$ Please guess $1+3+5+7+9+\\ldots +21=$______;\n$(2)$ Please guess $1+3+5+7+9+\\ldots +(2n-1)=\\_\\_\\_\\_\\_\\... | Explore the pattern, observe the following equations, and answer the questions.
$1=1^{2}$;
$1+3=4=2^{2}$;
$1+3+5=9=3^{2}$;
$1+3+5+7=16=4^{2}$;
$1+3+5+7+9=25=5^{2}$;
$\ldots $
$(1)$ Please guess $1+3+5+7+9+\ldots +21=$______;
$(2)$ Please guess $1+3+5+7+9+\ldots +(2n-1)=\_\_\_\_\_\_;(n$ is an integer and $n\geqslant 1)$... | "[\"12500\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107954, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "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:117916"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"skywork"
] | 88024e31aee12dfed0351f1596ff5d17646bb8486b4a33a6acc2137df2ffbf07 | [
{
"content": "We say an integer $n$ is naoish if $n \\geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$ and $90$ are naoish, whereas $9900$, $2009$ and $9$ are not. Nino expresses 2020 as a sum:\n \\[\n 2020=n_{1}+n_{2}+\\ldots+n_{k}\n \\]\n... | We say an integer $n$ is naoish if $n \geq 90$ and the second-to-last digit of $n$ (in decimal notation) is equal to $9$. For example, $10798$, $1999$ and $90$ are naoish, whereas $9900$, $2009$ and $9$ are not. Nino expresses 2020 as a sum:
\[
2020=n_{1}+n_{2}+\ldots+n_{k}
\]
where each of the $n_{j}$ ... | "[\"8\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107981, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117917"}] | {
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"any_reference_component_parses": true,
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"plain_numeric_reference": true,
"reference_components": 1,
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} |
[
"150"
] | [
"skywork"
] | bec50e2181c258d216000cf507223a40e5b806fc6d85da20897645f843828550 | [
{
"content": "1. Distribute 5 different books to 3 students, with each student getting at least one book. How many different ways are there to do this?",
"role": "user"
}
] | 1. Distribute 5 different books to 3 students, with each student getting at least one book. How many different ways are there to do this? | "[\"150\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107991, "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:117918"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"89"
] | [
"skywork"
] | 86d76300c087bed220beefbb751ebe2448e5a87c0b60bd32c6780589008501cf | [
{
"content": "If $n$ is the largest positive integer with $n^{2}<2018$ and $m$ is the smallest positive integer with $2018<m^{2}$, what is $m^{2}-n^{2}$ ?",
"role": "user"
}
] | If $n$ is the largest positive integer with $n^{2}<2018$ and $m$ is the smallest positive integer with $2018<m^{2}$, what is $m^{2}-n^{2}$ ? | "[\"89\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107986, "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:117919"}] | {
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} |
[
"4769280"
] | [
"skywork"
] | fa79452b76aedc0e774a66dc42b90debbfa4b3185be0c4992e2b2398e9f9aaad | [
{
"content": "7. There are 120 people. Any two people are either friends or not friends. A set of four people that contains exactly one pair of friends is called a \"weak quartet.\" Find the maximum number of \"weak quartets.\" (Provided by New Zealand)",
"role": "user"
}
] | 7. There are 120 people. Any two people are either friends or not friends. A set of four people that contains exactly one pair of friends is called a "weak quartet." Find the maximum number of "weak quartets." (Provided by New Zealand) | "[\"4769280\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 107980, "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:117920"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{256}"
] | [
"skywork"
] | 0e4ade67a61402d54a1b7ed1ea198157fef58ec3a0909a2f9c7659265b74fbc4 | [
{
"content": "7. Petya is playing a computer game called \"Pile of Stones.\" Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each ... | 7. Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, without violating ... | "[\"\\\\frac{1}{256}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107970, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "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:117921"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"(2,0)"
] | [
"skywork"
] | daefadbcbc5513ce0cf92872229249b4c050ac66d60fbd3c1f47d203c95f6549 | [
{
"content": "15. As shown in Figure 3, the upper semicircle with center at the origin and radius $r$ intersects the $y$-axis and the parabola $y^{2}=x$ at points $P$ and $M$, respectively. The line $PM$ intersects the $x$-axis at $Q$. Find the limit position of point $Q$ as the radius $r$ approaches zero.",
... | 15. As shown in Figure 3, the upper semicircle with center at the origin and radius $r$ intersects the $y$-axis and the parabola $y^{2}=x$ at points $P$ and $M$, respectively. The line $PM$ intersects the $x$-axis at $Q$. Find the limit position of point $Q$ as the radius $r$ approaches zero. | "[\"(2,0)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107963, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117922"}] | {
"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
} |
[
"10"
] | [
"skywork"
] | 6429a83e9e38ba979359eadb179687bb27002c376428c4bbe43b5fd773994683 | [
{
"content": "Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:\n\n[list]\n[*]A shakes hands with B, and B shakes hands with C, or\n[*]A doesn't shake hands with B, and B doesn't shake hands with C.\n... | Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a [i]trio[/i] if one of the following is true:
[list]
[*]A shakes hands with B, and B shakes hands with C, or
[*]A doesn't shake hands with B, and B doesn't shake hands with C.
[/list]
If we consider $(A... | "[\"10\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107958, "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:117923"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"k = 3"
] | [
"skywork"
] | 1c306a48f82350eef68c3a8bbc71cfa4973ffd3249c9c684c1a5a0c2495433f9 | [
{
"content": "Given is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$. Find the smallest $k$, such that each odd cycle has length at most $k$.",
"role": "user"
}
] | Given is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$. Find the smallest $k$, such that each odd cycle has length at most $k$. | "[\" k = 3 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107967, "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:117924"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"12"
] | [
"skywork"
] | 17f605fe931bbbf717746f524f4642558cdd14017388fb16b65a08d23790ccc2 | [
{
"content": "A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$?",
"role": "user"
}
] | A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$? | "[\"12\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107957, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117925"}] | {
"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-\\sqrt{3}"
] | [
"skywork"
] | 6156bc2c40172e81b9d4e7d91cc17260be93df5fc7d3ef977f2c423860f1518a | [
{
"content": "Define $a$ ? $=(a-1) /(a+1)$ for $a \\neq-1$. Determine all real values $N$ for which $(N ?)$ ?=\\tan 15.",
"role": "user"
}
] | Define $a$ ? $=(a-1) /(a+1)$ for $a \neq-1$. Determine all real values $N$ for which $(N ?)$ ?=\tan 15. | "[\"-2-\\\\sqrt{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107977, "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:117927"}] | {
"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
} |
[
"\\sqrt{3}"
] | [
"skywork"
] | f7d91dfca21b0d78c10672cc0d58f358fceea06ad51ecbfd1f3b4ce9b26c2a33 | [
{
"content": "11 If $\\log _{4}(x+2 y)+\\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\\qquad$ .",
"role": "user"
}
] | 11 If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\qquad$ . | "[\"\\\\sqrt{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108001, "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:117928"}] | {
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[
"26"
] | [
"skywork"
] | eebba61f29c1b0a802525408d3dbe0a39b940f3af0a27931533aaec254d4a8d4 | [
{
"content": "The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \\( (\\ \\ \\ ).",
"role": "user"
}
] | The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \( (\ \ \ ). | "[\"26\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107987, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117929"}] | {
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[
"97"
] | [
"skywork"
] | 314812bba8a904656b4fc99e19f3e42b1787cb49343a2722d2c55b610db52be2 | [
{
"content": "When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers all of which are nonprime?",
"role": "user"
}
] | When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers all of which are nonprime? | "[\"97\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 107989, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117930"}] | {
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[
"-5"
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"skywork"
] | 10569af544dc2c0640e27f4e520a0b2f95c5c76bd52334a80813c94aaf86d9dd | [
{
"content": "Solve the equations: \n① $3(x-1)^3 = 24$; \n② $(x-3)^2 = 64$.",
"role": "user"
}
] | Solve the equations:
① $3(x-1)^3 = 24$;
② $(x-3)^2 = 64$. | "[\"-5\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107994, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "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:117931"}] | {
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[
"f(x) = C + \\frac{x^2}{2a} - \\frac{x}{2}"
] | [
"skywork"
] | 12148d9a57e6bd633000ca06874d6c54825c545bd9329ae66825702ce27811e1 | [
{
"content": "Let $a\\in\\mathbb{R}-\\{0\\}$. Find all functions $f: \\mathbb{R}\\to\\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\\in\\mathbb{R}$.\r\n\r\n[i]Dan Schwartz[/i]",
"role": "user"
}
] | Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$.
[i]Dan Schwartz[/i] | "[\" f(x) = C + \\\\frac{x^2}{2a} - \\\\frac{x}{2} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107984, "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:117932"}] | {
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[
"(0;0),(\\frac{1}{2};0),(1;-1)"
] | [
"skywork"
] | 27d12988a65791db64fd06500053cf1b3f35a1415f29224edf028aad1dab1224 | [
{
"content": "6.245. Find the coefficients $m$ and $n$ of the quadratic trinomial $x^{2}+m x+n$, given that the remainders when it is divided by the binomials $x-m$ and $x-n$ are respectively $m$ and $\\boldsymbol{n}$.",
"role": "user"
}
] | 6.245. Find the coefficients $m$ and $n$ of the quadratic trinomial $x^{2}+m x+n$, given that the remainders when it is divided by the binomials $x-m$ and $x-n$ are respectively $m$ and $\boldsymbol{n}$. | "[\"(0;0),(\\\\frac{1}{2};0),(1;-1)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107995, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117933"}] | {
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"reference_components": 1,
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} |
[
"23"
] | [
"skywork"
] | 4c2350c044ebed1da4a715c40c51ad598d85b0c4c0fd8543dd5dd290102f421c | [
{
"content": "The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and again divided the cookie... | The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and again divided the cookies equally. This time, ... | "[\"23\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107979, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117934"}] | {
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[
"20"
] | [
"skywork"
] | b24087fffc1b8d6d5f39cd0ef7ed90a716de7414769fa195c28d1e0b7fe69da0 | [
{
"content": "8. An angle can be represented by two capital letters on its two sides and the letter of its vertex, such as $\\angle A O B$ symbol (\" $\\angle$ \" represents an angle), or it can also be represented by $\\angle O$ (when there is only one angle at the vertex). In the triangle $\\mathrm{ABC}$ belo... | 8. An angle can be represented by two capital letters on its two sides and the letter of its vertex, such as $\angle A O B$ symbol (" $\angle$ " represents an angle), or it can also be represented by $\angle O$ (when there is only one angle at the vertex). In the triangle $\mathrm{ABC}$ below, $\angle B A O=\angle C A ... | "[\"20\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107973, "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:117936"}] | {
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[
"82"
] | [
"skywork"
] | 8e96b1b42fc65b1baa12502aad356965b2d32106f4193d7779911919ecd97202 | [
{
"content": "1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\\{a_{n}\\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion\n$$\na_{n+1} = a_{n} + 2 \\left[\\sqrt{a_{n}}\\right].\n$$\n(1) If $... | 1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\{a_{n}\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion
$$
a_{n+1} = a_{n} + 2 \left[\sqrt{a_{n}}\right].
$$
(1) If $a = 8$, find the smallest posit... | "[\"82\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107971, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117937"}] | {
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[
"24"
] | [
"skywork"
] | 4d244d1ee7fd6bf99bd353c382978304f01c8b4050fd6b1a000a89236d96fd67 | [
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"content": "A list of five positive integers has all of the following properties:\n\n$\\bullet$ The only integer in the list that occurs more than once is $7,$ and it occurs three times.\n$\\bullet$ Its median is $10,$\n$\\bullet$ Its average (mean) is $11.$\n\nWhat is the largest possible integer that could ... | A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $7,$ and it occurs three times.
$\bullet$ Its median is $10,$
$\bullet$ Its average (mean) is $11.$
What is the largest possible integer that could appear in the list? | "[\"24\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 108004, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117938"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
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[
"32"
] | [
"skywork"
] | 68f61b1b859ffc25cd98e1ed380e76b3cf93eb05a0e7836a48175872d15af583 | [
{
"content": "4. (20 points) The ballroom in the palace of the thirtieth kingdom is a region on the plane, the coordinates of which satisfy the conditions $|x| \\leqslant 4,|y| \\leqslant 6$. How many identical parquet tiles, having the shape of a rectangle with sides 1.5 and 2, are needed to tile the floor of ... | 4. (20 points) The ballroom in the palace of the thirtieth kingdom is a region on the plane, the coordinates of which satisfy the conditions $|x| \leqslant 4,|y| \leqslant 6$. How many identical parquet tiles, having the shape of a rectangle with sides 1.5 and 2, are needed to tile the floor of the room? Tiling is cons... | "[\"32\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108000, "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:117939"}] | {
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[
"D"
] | [
"skywork"
] | 8b1421e14ee2206d0d29c57ac2fa3ddd57eabd6ac5114a39ce634374be4cd534 | [
{
"content": "17. The perimeter of an equilateral triangle is $1989 \\mathrm{~cm}$ longer than the perimeter of a square. The side length of the triangle is $d \\mathrm{~cm}$ longer than the side length of the square, and the perimeter of the square is greater than 0. The number of positive integer values that ... | 17. The perimeter of an equilateral triangle is $1989 \mathrm{~cm}$ longer than the perimeter of a square. The side length of the triangle is $d \mathrm{~cm}$ longer than the side length of the square, and the perimeter of the square is greater than 0. The number of positive integer values that $d$ cannot take is $\qqu... | "[\"D\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 107983, "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:117940"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"5"
] | [
"skywork"
] | 191747200934eb859247c33c10b61d972b06c660395b2c706aa7850f672fcee9 | [
{
"content": "How many different lists $a, b, c, d$ of distinct odd positive integers with $a<b<c<d$ have the property that $a+b+c+d=24$ ?",
"role": "user"
}
] | How many different lists $a, b, c, d$ of distinct odd positive integers with $a<b<c<d$ have the property that $a+b+c+d=24$ ? | "[\"5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108007, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "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:117941"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"861"
] | [
"skywork"
] | e828ba37a551760e05611cb8fbfffdf57f8fab7231045cd41b8ae1ea8794fcb6 | [
{
"content": "Given $$a_{n}= \\frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\\{a_n\\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\\{b_n\\}$. Find the value of $b_{21}$.",
"role": "user"
}
] | Given $$a_{n}= \frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\{a_n\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\{b_n\}$. Find the value of $b_{21}$. | "[\"861\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107997, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117942"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"55"
] | [
"skywork"
] | 53533c09674afa5f6ecc3ad1387dab5fc982c4a41edd3049eaef56c300c6415b | [
{
"content": "2. A school coach decided to reward 12 students who ran the distance in the best time. Each of them must be awarded a \"gold\", \"silver\", or \"bronze\" medal. All three types of medals must be used, and the one who finished earlier cannot be awarded a less valuable medal than the one who finishe... | 2. A school coach decided to reward 12 students who ran the distance in the best time. Each of them must be awarded a "gold", "silver", or "bronze" medal. All three types of medals must be used, and the one who finished earlier cannot be awarded a less valuable medal than the one who finished later.
In how many ways c... | "[\"55\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107998, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "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:117943"}] | {
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"plain_numeric_reference": true,
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[
"6"
] | [
"skywork"
] | f6babd0e17c843e1dc4af67d35fdec12f7271cff904b572ae67c2ae987dad4b9 | [
{
"content": "PROBLEM 1. Consider the distinct rational numbers $a_{1}, a_{2}, \\ldots, a_{n}$, such that among any four of them, there are two whose product is -1.\n\na) Give an example of 6 such numbers;\n\nb) Find the maximum value of $n$.",
"role": "user"
}
] | PROBLEM 1. Consider the distinct rational numbers $a_{1}, a_{2}, \ldots, a_{n}$, such that among any four of them, there are two whose product is -1.
a) Give an example of 6 such numbers;
b) Find the maximum value of $n$. | "[\"6\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 107999, "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:117944"}] | {
"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
} |
[
"-60"
] | [
"skywork"
] | 881d77bfa1fd688269964f550a74f100d28532648656dbc0957f8f41f779115b | [
{
"content": "Simplify and find the value of:<br/>$(1)$ If $a=2$ and $b=-1$, find the value of $(3{a^2}b+\\frac{1}{4}a{b^2})-(\\frac{3}{4}a{b^2}-{a^2}b)$.<br/>$(2)$ If the value of the algebraic expression $(2x^{2}+ax-y+6)-(2bx^{2}-3x+5y-1)$ is independent of the variable $x$, find the value of the algebraic ex... | Simplify and find the value of:<br/>$(1)$ If $a=2$ and $b=-1$, find the value of $(3{a^2}b+\frac{1}{4}a{b^2})-(\frac{3}{4}a{b^2}-{a^2}b)$.<br/>$(2)$ If the value of the algebraic expression $(2x^{2}+ax-y+6)-(2bx^{2}-3x+5y-1)$ is independent of the variable $x$, find the value of the algebraic expression $5ab^{2}-[a^{2}... | "[\"-60\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 107993, "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:117946"}] | {
"all_reference_components_parse": true,
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"reference_components": 1,
"reference_present": true
} |
[
"x\\in(-\\infty;-1)\\cup(0;1)\\cup(1;+\\infty)"
] | [
"skywork"
] | fdc40ffe1e3fd8c446fd6f8a0ae06e89b34f548ec88693d5f9828520a0e0820a | [
{
"content": "$9.249 \\log _{a} \\frac{1+\\log _{a}^{2} x}{1-\\log _{a} x}<0$.\n\nTranslate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n$9.249 \\log _{a} \\frac{1+\\log _{a}^{2} x}{1-\\log _{a} x}<0$.",
"role": "user"
}
] | $9.249 \log _{a} \frac{1+\log _{a}^{2} x}{1-\log _{a} x}<0$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
$9.249 \log _{a} \frac{1+\log _{a}^{2} x}{1-\log _{a} x}<0$. | "[\"x\\\\in(-\\\\infty;-1)\\\\cup(0;1)\\\\cup(1;+\\\\infty)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108006, "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:117947"}] | {
"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
} |
[
"\\binom{N}{3}"
] | [
"skywork"
] | 888ce43f2b65a6bf76825812558435c8da3c392fcfdd709ca2b28043350f2fa2 | [
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"content": "$N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would... | $N$ oligarchs built a country with $N$ cities with each one of them owning one city. In addition, each oligarch built some roads such that the maximal amount of roads an oligarch can build between two cities is $1$ (note that there can be more than $1$ road going through two cities, but they would belong to different o... | "[\"\\\\binom{N}{3}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 107974, "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:117948"}] | {
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[
"1"
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"skywork"
] | 0f0e52b01d6893c3b3250767ce012e5e23782b4335ce743da5247316dc60a890 | [
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"content": "[ Fermat's Little Theorem ]\n\nFind the remainder when $2^{100}$ is divided by 101.\n\n#",
"role": "user"
}
] | [ Fermat's Little Theorem ]
Find the remainder when $2^{100}$ is divided by 101.
# | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108016, "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:117949"}] | {
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[
"\\arccos\\frac{5}{8}"
] | [
"skywork"
] | 778b0ef94de27bb5b024e4f9aa95f80ebab57574f53a0b42f9480d658b32f8ec | [
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"content": "[ Completing a Tetrahedron to a Parallelepiped ]\n\nThe cosine of the angle between the skew lines $A B$ and $C D$ is $\\frac{\\sqrt{35}}{10}$. Points $E$ and $F$ are the midpoints of segments $A B$ and $C D$ respectively, and the line $E F$ is perpendicular to the lines $A B$ and $C D$. Find the ... | [ Completing a Tetrahedron to a Parallelepiped ]
The cosine of the angle between the skew lines $A B$ and $C D$ is $\frac{\sqrt{35}}{10}$. Points $E$ and $F$ are the midpoints of segments $A B$ and $C D$ respectively, and the line $E F$ is perpendicular to the lines $A B$ and $C D$. Find the angle $A C B$, given that ... | "[\"\\\\arccos\\\\frac{5}{8}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108005, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117950"}] | {
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[
"\\frac{1}{3}\\piS\\frac{\\sqrt{2S\\sin2\\alpha}}{\\sin^{2}2\\alpha\\cos^{3}\\alpha}"
] | [
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"content": "12.098. A cone is inscribed in a sphere. The area of the axial section of the cone is $S$, and the angle between the height and the slant height is $\\alpha$. Find the volume of the sphere.",
"role": "user"
}
] | 12.098. A cone is inscribed in a sphere. The area of the axial section of the cone is $S$, and the angle between the height and the slant height is $\alpha$. Find the volume of the sphere. | "[\"\\\\frac{1}{3}\\\\piS\\\\frac{\\\\sqrt{2S\\\\sin2\\\\alpha}}{\\\\sin^{2}2\\\\alpha\\\\cos^{3}\\\\alpha}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108002, "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:117951"}] | {
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[
"300"
] | [
"skywork"
] | 2a9ba7fc3e9e1325c13120df8213c9b3da53a1c4acebc4ea06b96af1ddf5f522 | [
{
"content": "Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\\$180$. On Wednesday, she sells twe... | Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts... | "[\"300\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108012, "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:117952"}] | {
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[
"x\\in[3-6\\log_{2}17;1]\\cup(3;+\\infty)"
] | [
"skywork"
] | 82b0fad8dabcca61e9051d126127421a022ac347df65e7dcb29c723cf3a61fd7 | [
{
"content": "3. Solve the inequality $17^{\\frac{5 x-3}{3-x}} \\cdot 2^{3-x} \\leq 68$.\n\n---\n\nThe provided text has been translated into English while preserving the original formatting and line breaks.",
"role": "user"
}
] | 3. Solve the inequality $17^{\frac{5 x-3}{3-x}} \cdot 2^{3-x} \leq 68$.
---
The provided text has been translated into English while preserving the original formatting and line breaks. | "[\"x\\\\in[3-6\\\\log_{2}17;1]\\\\cup(3;+\\\\infty)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108028, "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:117953"}] | {
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[
"1"
] | [
"skywork"
] | 0fda2ae221e09eacab13247fbf2d757341f1f843cd72dbcaa89205a500c8ee1a | [
{
"content": "What is the greatest amount of rooks that can be placed on an $n\\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them.\n\n[I]Proposed by D. Karpov[/i]",
"... | What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them.
[I]Proposed by D. Karpov[/i] | "[\"1\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108011, "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:117955"}] | {
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[
"21"
] | [
"skywork"
] | 3f682983b1b4f6e10cec5a5914bafe05d08981affcdf59a112ef8602a789aba8 | [
{
"content": "Lisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles?",
"role": "user"
}
] | Lisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles? | "[\"21\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108036, "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:117956"}] | {
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[
"D"
] | [
"skywork"
] | bb4ffcf02d5f79e4561f2fe4f2a5f5f033381d5a0090511a490bdfbfb5bebb23 | [
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"content": "1.21 Real numbers $x, y, z$ satisfy $x+y+z=x y z$, then the following statements I. If two of $x, y, z$ are opposite numbers, then the third number must be 0. II. If one of $x, y, z$ is 0, then the other two must be opposite numbers.\nIII. $z=\\frac{x+y}{x y-1}$.\n(A) Only I is true, II and III ar... | 1.21 Real numbers $x, y, z$ satisfy $x+y+z=x y z$, then the following statements I. If two of $x, y, z$ are opposite numbers, then the third number must be 0. II. If one of $x, y, z$ is 0, then the other two must be opposite numbers.
III. $z=\frac{x+y}{x y-1}$.
(A) Only I is true, II and III are not true.
(B) Only I is... | "[\"D\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108014, "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:117957"}] | {
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[
"93"
] | [
"skywork"
] | c9f7f8865cb710958461524eb82777246611ea5e06e67e14b3b8e33a6f5733b7 | [
{
"content": "1. For every natural number $n$ with 3 decimal digits (so the first digit is not zero), we consider the number $n_{0}$ obtained from $n$ by removing its digits that are equal to zero. For example, if $n=205$ then $n_{0}=25$.\n\nDetermine the number of integers $n$ with three digits for which $n_{0... | 1. For every natural number $n$ with 3 decimal digits (so the first digit is not zero), we consider the number $n_{0}$ obtained from $n$ by removing its digits that are equal to zero. For example, if $n=205$ then $n_{0}=25$.
Determine the number of integers $n$ with three digits for which $n_{0}$ is a divisor of $n$ d... | "[\"93\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108010, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117958"}] | {
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[
"14\\mathrm{~}"
] | [
"skywork"
] | 9f059a37af08eb5ea7ef21a588f8097ed46542ab2afbb0c89bc50e217898dfc9 | [
{
"content": "## Task 2\n\nIn sports, all children are trying very hard. Horst throws the ball 18 meters far. Ines only reaches half of that. Claudia throws the ball 5 meters further than Ines.\n\na) How many meters does Ines reach?\n\nb) How far does Claudia throw her ball?",
"role": "user"
}
] | ## Task 2
In sports, all children are trying very hard. Horst throws the ball 18 meters far. Ines only reaches half of that. Claudia throws the ball 5 meters further than Ines.
a) How many meters does Ines reach?
b) How far does Claudia throw her ball? | "[\"14\\\\mathrm{~}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108020, "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:117959"}] | {
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[
"8"
] | [
"skywork"
] | 1010f8239d4bde83cf2b8ba731dddf56d9195b34755a6c01178be83ed0f4d3a2 | [
{
"content": "In the four term sequence $0.001,0.02,0.4, x$, each term after the first is found by multiplying the previous term by the same number. What is the value of $x$ ?\n(A) 0.8\n(B) 8\n(C) 80\n(D) 8.8\n(E) 0.08",
"role": "user"
}
] | In the four term sequence $0.001,0.02,0.4, x$, each term after the first is found by multiplying the previous term by the same number. What is the value of $x$ ?
(A) 0.8
(B) 8
(C) 80
(D) 8.8
(E) 0.08 | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108030, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117960"}] | {
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[
"6^{n-2}"
] | [
"skywork"
] | 710168ed1d30f197464070693910bb451423b516a481f883b22d149c3dcf3f6e | [
{
"content": "Five. (20 points) How many $n$-digit numbers can be formed from the digits $1,2, \\cdots, 6$ such that $1$ and $6$ are adjacent?",
"role": "user"
}
] | Five. (20 points) How many $n$-digit numbers can be formed from the digits $1,2, \cdots, 6$ such that $1$ and $6$ are adjacent? | "[\"6^{n-2}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 108018, "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:117961"}] | {
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[
"30\\sqrt{3}"
] | [
"skywork"
] | 8999f123ea4f498b410f0f7c9ec38df73682a257d29ca9d644910765ee23943e | [
{
"content": "Point \\( M \\) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \\( M \\) to the lines containing the other sides of the hexagon.",
"role": "user"
}
] | Point \( M \) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \( M \) to the lines containing the other sides of the hexagon. | "[\"30\\\\sqrt{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108032, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "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:117962"}] | {
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[
"11"
] | [
"skywork"
] | e6fb5680ce2a28f2c58a4883763f479f22214a4cc35c027b5a152d3e23619af3 | [
{
"content": "3. A dish contains 100 candies. Juan removes candies from the dish each day and no candies are added to the dish. On day 1, Juan removes 1 candy. On day 2, Juan removes 2 candies. On each day that follows, Juan removes 1 more candy than he removed on the previous day. After day $n$, Juan has remov... | 3. A dish contains 100 candies. Juan removes candies from the dish each day and no candies are added to the dish. On day 1, Juan removes 1 candy. On day 2, Juan removes 2 candies. On each day that follows, Juan removes 1 more candy than he removed on the previous day. After day $n$, Juan has removed a total of at least... | "[\"11\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108024, "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:117963"}] | {
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[
"1"
] | [
"skywork"
] | 568464d9bb071c747565445565ea1d99befff60be7a8028b6e7bb0e4f2313363 | [
{
"content": "## Subject IV. (20 points)\n\nConsider the sequences $\\left(a_{n}\\right)_{n>1}$ and $\\left(b_{n}\\right)_{n>1}$, where: $a_{n}=\\sum_{k=1}^{n} \\frac{k+2}{k!+(k+1)!+(k+2)!}$ and $b_{n}=\\sum_{k=1}^{n} k!(k+2)$\n\nRequired:\na) $\\lim _{n \\rightarrow \\infty} a_{n}$;\nb) $\\lim _{n \\rightarrow... | ## Subject IV. (20 points)
Consider the sequences $\left(a_{n}\right)_{n>1}$ and $\left(b_{n}\right)_{n>1}$, where: $a_{n}=\sum_{k=1}^{n} \frac{k+2}{k!+(k+1)!+(k+2)!}$ and $b_{n}=\sum_{k=1}^{n} k!(k+2)$
Required:
a) $\lim _{n \rightarrow \infty} a_{n}$;
b) $\lim _{n \rightarrow \infty}\left(2 a_{n}\right)^{b_{n}}$.
... | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108017, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117964"}] | {
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[
"366"
] | [
"skywork"
] | bc4195fadd45670e3a563f9781401e8c75309f2ba94e0d9c211ae875f0868057 | [
{
"content": "Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \\leq a, c \\leq 59$ and $1 \\leq b, d \\leq 5$.",
"role"... | Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \leq a, c \leq 59$ and $1 \leq b, d \leq 5$. | "[\"366\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108019, "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:117965"}] | {
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[
"2.5"
] | [
"skywork"
] | 0c947eda77fa4609ed8cd518c1121d3bc410ea91b5fd78b658c78052e34342e4 | [
{
"content": "At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal.",
"role": "user"
}
... | At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal. | "[\"2.5\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108048, "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:117966"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
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[
"9"
] | [
"skywork"
] | 08d9c15b9d9364903e2efc8e21066ebcba44e0d90b06f6b74d0942531f3eed4b | [
{
"content": "6. If $a>b>c>d$, and $\\frac{1}{a-b}+\\frac{1}{b-c}+\\frac{1}{c-d} \\geqslant \\frac{n}{a-d}$, then the maximum value of $n$ is ( ).\nA. 12\nB. 9\nC. 6\nD. 3",
"role": "user"
}
] | 6. If $a>b>c>d$, and $\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-d} \geqslant \frac{n}{a-d}$, then the maximum value of $n$ is ( ).
A. 12
B. 9
C. 6
D. 3 | "[\"9\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108042, "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:117967"}] | {
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"reference_components": 1,
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} |
[
"6"
] | [
"skywork"
] | fbf4f14de61d347dc22ad67e84b34bf6cdd507a3df4820889815ae6034c5ad17 | [
{
"content": "One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have?",
"role": "user"
}
] | One interior angle of a convex polygon is 160 degrees. The rest of the interior angles of the polygon are each 112 degrees. How many sides does the polygon have? | "[\"6\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108046, "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:117968"}] | {
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"reference_components": 1,
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} |
[
"f(n)=n"
] | [
"skywork"
] | f420351b6931db47e7e8e31439dad3731951f4e93c99f74c9b587075eacc02bb | [
{
"content": "Find all functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for all natural numbers $a$ and $b$:\n\n$$\nf(f(a)+f(b))=a+b\n$$",
"role": "user"
}
] | Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all natural numbers $a$ and $b$:
$$
f(f(a)+f(b))=a+b
$$ | "[\"f(n)=n\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108026, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117969"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | 1e87702a14f09cd1100b0ffdccede5010f2dfd5cc8a5cd95be80fe013e276a99 | [
{
"content": "When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3?",
"role": "user"
}
] | When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3? | "[\"1\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108056, "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:117970"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{6}"
] | [
"skywork"
] | 39002e2196ecdad8794ea3fc8c9a89538ea49b24cff0fe6a53001aeaf4a44723 | [
{
"content": "Let $n\\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\\ X_2,\\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once ... | Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is n... | "[\"\\\\frac{1}{6}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108009, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "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:117971"}] | {
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} |
[
"\\frac{b+1}{b-2a}"
] | [
"skywork"
] | ccb546a0d1a554b48b4059183addbf186c1cab4779b7a45f4ea0f5d5a915d897 | [
{
"content": "2.047. $\\frac{\\frac{a-b}{2a-b}-\\frac{a^{2}+b^{2}+a}{2a^{2}+ab-b^{2}}}{\\left(4b^{4}+4ab^{2}+a^{2}\\right):\\left(2b^{2}+a\\right)} \\cdot\\left(b^{2}+b+ab+a\\right)$.",
"role": "user"
}
] | 2.047. $\frac{\frac{a-b}{2a-b}-\frac{a^{2}+b^{2}+a}{2a^{2}+ab-b^{2}}}{\left(4b^{4}+4ab^{2}+a^{2}\right):\left(2b^{2}+a\right)} \cdot\left(b^{2}+b+ab+a\right)$. | "[\"\\\\frac{b+1}{b-2a}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108033, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:117972"}] | {
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"reference_components": 1,
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} |
[
"108"
] | [
"skywork"
] | 60cd245be43f83a78806fecd0532bd33055054ee4f22254e2f4e38310f6f4c09 | [
{
"content": "7. Given the real-coefficient equation $a x^{3}-x^{2}+b x-1=0$ has three positive real roots. Then\n$$\nP=\\frac{5 a^{2}-6 a b+3}{a^{3}(b-a)}\n$$\n\nthe minimum value of $P$ is",
"role": "user"
}
] | 7. Given the real-coefficient equation $a x^{3}-x^{2}+b x-1=0$ has three positive real roots. Then
$$
P=\frac{5 a^{2}-6 a b+3}{a^{3}(b-a)}
$$
the minimum value of $P$ is | "[\"108\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 108015, "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:117973"}] | {
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"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"252"
] | [
"skywork"
] | 95ce7e502ab6db31f8aa7cfe84e2da0b1226c1ac67f27e2a97c440686e0c5d5b | [
{
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate.",
"role": "user"
}
] | A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. | "[\"252\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108031, "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:117974"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"\\frac{5\\pi}{6}"
] | [
"skywork"
] | 67d28b36059caf522421378ada6eb997181a6e2a27a5e3cb7b4fce50d1695431 | [
{
"content": "Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\\frac{\\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajecto... | Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $... | "[\"\\\\frac{5\\\\pi}{6}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108034, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117975"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"7.94403"
] | [
"skywork"
] | 86e7c0b56cf16d38b4c19c756949d3e9b373308ca80d4d43b3e67f0afdff9e58 | [
{
"content": "Let $C$ be the left vertex of the ellipse $\\frac{x^2}8+\\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$.\n\n(i) Compute the maximum of $|CA|+|CB|$.\n(ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and... | Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$.
(i) Compute the maximum of $|CA|+|CB|$.
(ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the ... | "[\" 7.94403 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108025, "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:117976"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"\\theta \\in\\left(\\frac{\\pi}{4}, \\frac{\\pi}{2}\\right]"
] | [
"skywork"
] | 1f86af849c65483b432e371eec36163a3ee33ee200b74cd26198aa03fef67070 | [
{
"content": "2. If $\\sqrt{\\cos \\theta}-\\sqrt{\\sin \\theta}<5\\left(\\sin ^{3} \\theta-\\cos ^{3} \\theta\\right)$, $\\theta \\in[0,2 \\pi)$, then the range of values for $\\theta$ is . $\\qquad$",
"role": "user"
}
] | 2. If $\sqrt{\cos \theta}-\sqrt{\sin \theta}<5\left(\sin ^{3} \theta-\cos ^{3} \theta\right)$, $\theta \in[0,2 \pi)$, then the range of values for $\theta$ is . $\qquad$ | "[\"\\\\theta \\\\in\\\\left(\\\\frac{\\\\pi}{4}, \\\\frac{\\\\pi}{2}\\\\right]\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 108039, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117977"}] | {
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} |
[
"(1, 1, 1)"
] | [
"skywork"
] | 87990e51c0e14c958cb6ef6110a942a075dc4865303d91c121e0325f3a514bbb | [
{
"content": "Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\\frac{1}{y},y+\\frac{1}{z}$ and $z+\\frac{1}{x}$ are integers.",
"role": "user"
}
] | Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\frac{1}{y},y+\frac{1}{z}$ and $z+\frac{1}{x}$ are integers. | "[\" (1, 1, 1) \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108044, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117978"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"20"
] | [
"skywork"
] | f6dc6df8fbb0f5cfa8d6c3002ee88d2ea70bf7bf230d1eb3fd20d005153cb00e | [
{
"content": "When $x=2$, the expression $2 x^{2}+3 x^{2}$ equals\n(A) 14\n(B) 10\n(C) 12\n(D) 22\n(E) 20",
"role": "user"
}
] | When $x=2$, the expression $2 x^{2}+3 x^{2}$ equals
(A) 14
(B) 10
(C) 12
(D) 22
(E) 20 | "[\"20\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108052, "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:117979"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"\\frac{4}{9}"
] | [
"skywork"
] | 318c72dabecbcf66a1379cf4aee083711ce6c138ed72c1c26bef1aee2cc10666 | [
{
"content": "## Task B-2.3.\n\nOn the side $\\overline{A B}$ of triangle $A B C$, a point $E$ is given such that $|A E|=1$ and $|E B|=2$. Let point $D$ be on the side $\\overline{A C}$ such that the segment $\\overline{D E}$ is parallel to the segment $\\overline{B C}$, and point $F$ be on the side $\\overline... | ## Task B-2.3.
On the side $\overline{A B}$ of triangle $A B C$, a point $E$ is given such that $|A E|=1$ and $|E B|=2$. Let point $D$ be on the side $\overline{A C}$ such that the segment $\overline{D E}$ is parallel to the segment $\overline{B C}$, and point $F$ be on the side $\overline{B C}$ such that the segment ... | "[\"\\\\frac{4}{9}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108040, "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:117980"}] | {
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"any_reference_component_parses": true,
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"reference_components": 1,
"reference_present": true
} |
[
"624"
] | [
"skywork"
] | 39041151fba013b084c699c616f5f0fc8ea2d0d920c9f7bbb9c892508c455cd6 | [
{
"content": "5. How many natural numbers less than 100000 are divisible by 4 and in whose decimal representation only the digits $0,1,2,3$ and 5 participate? (Digits can repeat and not all of them need to appear in the representation of such a number.)\n\n## Second Grade - B category",
"role": "user"
}
] | 5. How many natural numbers less than 100000 are divisible by 4 and in whose decimal representation only the digits $0,1,2,3$ and 5 participate? (Digits can repeat and not all of them need to appear in the representation of such a number.)
## Second Grade - B category | "[\"624\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108043, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117981"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-1"
] | [
"skywork"
] | 20d7eba67556d35eaf268eb89b6cfa1ae67b629b722504631040f1513b0f25e3 | [
{
"content": "If the positive real numbers \\(a\\) and \\(b\\) satisfy \\(\\frac{1}{a} + \\frac{1}{b} \\leq 2 \\sqrt{2}\\) and \\((a - b)^2 = 4 (ab)^3\\), then \\(\\log_a b =\\) ?",
"role": "user"
}
] | If the positive real numbers \(a\) and \(b\) satisfy \(\frac{1}{a} + \frac{1}{b} \leq 2 \sqrt{2}\) and \((a - b)^2 = 4 (ab)^3\), then \(\log_a b =\) ? | "[\"-1\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108029, "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:117982"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"108, 135, 180, 117"
] | [
"skywork"
] | d3e3a816fd34b931d3112654378c1b787d526a0da029f86acb701b789af2bcb1 | [
{
"content": "1.37 Four three-digit numbers with the same first digit are distinct and have the property: the sum of these numbers is divisible by any 3 of them. Find these 4 numbers.\n\nThe text above is translated into English, preserving the original text's line breaks and format.",
"role": "user"
}
] | 1.37 Four three-digit numbers with the same first digit are distinct and have the property: the sum of these numbers is divisible by any 3 of them. Find these 4 numbers.
The text above is translated into English, preserving the original text's line breaks and format. | "[\"108, 135, 180, 117\"]" | train-math-numinamath1.5_inequalities | [{"dataset": "skywork", "extra_info": {"index": 108023, "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:117983"}] | {
"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
} |
[
"x\\in(-5;-2)\\cup(2;3)\\cup(3;5)"
] | [
"skywork"
] | 1d78aae642e7de1e5b09ccba256d78b5e1a0c46caebfeb9feb18d62a3a63fb4d | [
{
"content": "$9.275 \\frac{x^{2}-7|x|+10}{x^{2}-6 x+9}<0$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.\n\n$9.275 \\frac{x^{2}-7|x|+10}{x^{2}-6 x+9}<0$.",
"role": "user"
}
] | $9.275 \frac{x^{2}-7|x|+10}{x^{2}-6 x+9}<0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$9.275 \frac{x^{2}-7|x|+10}{x^{2}-6 x+9}<0$. | "[\"x\\\\in(-5;-2)\\\\cup(2;3)\\\\cup(3;5)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108060, "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:117984"}] | {
"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
} |
[
"289/259"
] | [
"skywork"
] | e8274adbbe789528fe4c1faac196407eeb05e21dfc217d6b87fa12d65ce9cc18 | [
{
"content": "The polynomial \\( f(x)=x^{2007}+17 x^{2006}+1 \\) has distinct zeroes \\( r_{1}, \\ldots, r_{2007} \\). A polynomial \\( P \\) of degree 2007 has the property that \\( P\\left(r_{j}+\\frac{1}{r_{j}}\\right)=0 \\) for \\( j=1, \\ldots, 2007 \\). Determine the value of \\( P(1) / P(-1) \\).",
"... | The polynomial \( f(x)=x^{2007}+17 x^{2006}+1 \) has distinct zeroes \( r_{1}, \ldots, r_{2007} \). A polynomial \( P \) of degree 2007 has the property that \( P\left(r_{j}+\frac{1}{r_{j}}\right)=0 \) for \( j=1, \ldots, 2007 \). Determine the value of \( P(1) / P(-1) \). | "[\"289/259\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108027, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "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:117985"}] | {
"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
} |
[
"47"
] | [
"skywork"
] | d38a1560effbb237b3ba37244904d6c413741e158610fa61f9bde45fff264ba2 | [
{
"content": "## Task 1\n\nParents of class 2a want to host guests for the festival. 8 parents each host 3 guests, 7 parents each host 2 guests, and 9 parents each host 1 guest.\n\nHow many guests will be hosted by the parents of class 2a?",
"role": "user"
}
] | ## Task 1
Parents of class 2a want to host guests for the festival. 8 parents each host 3 guests, 7 parents each host 2 guests, and 9 parents each host 1 guest.
How many guests will be hosted by the parents of class 2a? | "[\"47\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108067, "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:117986"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"10"
] | [
"skywork"
] | 7c17c155c2d33368cd8e81184be7aa97f9e213a4af745018b5ae5f8c8f32e43b | [
{
"content": "Given that in $\\triangle ABC$, $AB=4$, $AC=6$, $BC= \\sqrt{7}$, and the center of its circumcircle is $O$, find $\\overset{⇀}{AO}· \\overset{⇀}{BC} =$ ___.",
"role": "user"
}
] | Given that in $\triangle ABC$, $AB=4$, $AC=6$, $BC= \sqrt{7}$, and the center of its circumcircle is $O$, find $\overset{⇀}{AO}· \overset{⇀}{BC} =$ ___. | "[\"10\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108055, "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:117987"}] | {
"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,-2,-9)"
] | [
"skywork"
] | 5f87e1d3b19fe12937ddc453f701586e35673aa16a92711b45a6e6c07d74ee05 | [
{
"content": "The polynomial $f(x) = x^3 + x^2 + 2x + 3$ has three distinct roots. Let $g(x) = x^3+bx^2+cx+d$ be a cubic polynomial with leading coefficient $1$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Find the ordered triple $(b,c,d)$.",
"role": "user"
}
] | The polynomial $f(x) = x^3 + x^2 + 2x + 3$ has three distinct roots. Let $g(x) = x^3+bx^2+cx+d$ be a cubic polynomial with leading coefficient $1$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Find the ordered triple $(b,c,d)$. | "[\"(3,-2,-9)\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108050, "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:117988"}] | {
"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
} |
[
"5"
] | [
"skywork"
] | 73cf6b8965d172a83424fe62e6b0b9ccff160f74c88fe98dfd3098b80518da29 | [
{
"content": "I3.1 若 $\\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, 求 $\\mathrm{f}\\left(x^{5}\\right)$ 除以 $\\mathrm{f}(x)$ 的餘值 $a$ 。\nIf $\\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, determine the remainder $a$ of $\\mathrm{f}\\left(x^{5}\\right)$ divided by $\\mathrm{f}(x)$.",
"role": "user"
}
] | I3.1 若 $\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, 求 $\mathrm{f}\left(x^{5}\right)$ 除以 $\mathrm{f}(x)$ 的餘值 $a$ 。
If $\mathrm{f}(x)=x^{4}+x^{3}+x^{2}+x+1$, determine the remainder $a$ of $\mathrm{f}\left(x^{5}\right)$ divided by $\mathrm{f}(x)$. | "[\"5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 108059, "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:117989"}] | {
"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
} |
[
"a_1 a_2 \\cdots a_n \\geq m^2 M^{n-2}"
] | [
"skywork"
] | c6cac37ef6b9f380b31d91a858b4209d0a7e707cffcc0272f1b5985cb958d07b | [
{
"content": "Let $n \\geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \\leq i,j,k \\leq n$, if $a_i \\leq a_j \\leq a_k$ then $a_ia_k \\leq a_j^2$. Show that\n\n\\[ a_1a_2 \\... | Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that
\[ a_1a_2 \cdots a_n \geq m^2M^{n-2} \]
an... | "[\" a_1 a_2 \\\\cdots a_n \\\\geq m^2 M^{n-2} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 108054, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 7, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:117990"}] | {
"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
} |
[
"450\\sqrt{2}"
] | [
"skywork"
] | 3525101c911ec4590931bf52b64dbb0953b59bb1162da6b0613d006a7875619f | [
{
"content": "Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=10$. Find the volume of the tetrahedron.",
"role": "user"
}
] | Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=10$. Find the volume of the tetrahedron. | "[\"450\\\\sqrt{2}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 108064, "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:117992"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{14}{3}\\sqrt{7}"
] | [
"skywork"
] | 8800bbb2a560957abfbcab8bdeeb763e977e5c287597814729882b5814969cd5 | [
{
"content": "Given a parabola \\( y^2 = 6x \\) with two variable points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), where \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\). The perpendicular bisector of segment \\( AB \\) intersects the x-axis at point \\( C \\). Find the maximum area of triangle \\( \\triangle... | Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \). | "[\"\\\\frac{14}{3}\\\\sqrt{7}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108021, "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:117993"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"-16"
] | [
"skywork"
] | 180a71b419c778b17ec114f534084b735dedfbf4284a344f762db6f9cc1afbca | [
{
"content": "If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$.",
"role": "user"
}
] | If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. | "[\"-16\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 108038, "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:117994"}] | {
"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}{3} \\leqslant a^{2}-a b+b^{2} \\leqslant 3"
] | [
"skywork"
] | 2b74b67d5f72854a964e2b393f0f1d7adc3ce1bafe0f221cd1fed0d83f126255 | [
{
"content": "11. (Full marks 12 points) Given real numbers $a$ and $b$ satisfy $a^{2} + ab + b^{2} = 1$. Find the range of $a^{2} - ab + b^{2}$.",
"role": "user"
}
] | 11. (Full marks 12 points) Given real numbers $a$ and $b$ satisfy $a^{2} + ab + b^{2} = 1$. Find the range of $a^{2} - ab + b^{2}$. | "[\"\\\\frac{1}{3} \\\\leqslant a^{2}-a b+b^{2} \\\\leqslant 3\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 108072, "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:117995"}] | {
"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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