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 |
|---|---|---|---|---|---|---|---|---|
[
"n \\geq 3"
] | [
"skywork"
] | 93102afdac4f98f955ca95502e4b49185bec29fbf4a11d9ccfb626c07071fc81 | [
{
"content": "Determine all $n$ for which the system with of equations can be solved in $\\mathbb{R}$:\r\n\r\n\\[\\sum^{n}_{k=1} x_k = 27\\]\r\n\r\nand\r\n\r\n\\[\\prod^{n}_{k=1} x_k = \\left( \\frac{3}{2} \\right)^{24}.\\]",
"role": "user"
}
] | Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$:
\[\sum^{n}_{k=1} x_k = 27\]
and
\[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\] | "[\" n \\\\geq 3 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1509, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15504"}] | {
"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"
] | 58ef7c6536cec00a6c91530aa195a456c46024b7c37653e266c77eca61999149 | [
{
"content": "Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\\frac{4}{5}$ as $N$... | Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is th... | "[\"12\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1481, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "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:15505"}, {"dataset": "skywork", "extra_in... | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0.37"
] | [
"skywork"
] | d141df4536e4755f05c846cd27a1bf576b7b2139112248f60700bdfb33257dde | [
{
"content": "Example 6.28. A radio system, having 1000 elements (with a failure rate $\\lambda_{i}=10^{-6}$ failures/hour), has passed testing and has been accepted by the customer. It is required to determine the probability of failure-free operation of the system in the interval $t_{1}<\\left(t=t_{1}+\\Delta... | Example 6.28. A radio system, having 1000 elements (with a failure rate $\lambda_{i}=10^{-6}$ failures/hour), has passed testing and has been accepted by the customer. It is required to determine the probability of failure-free operation of the system in the interval $t_{1}<\left(t=t_{1}+\Delta t\right)<t_{2}$, where $... | "[\"0.37\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1511, "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:15506"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{10}{29}"
] | [
"skywork"
] | 613183ac86ae5980e02fa412c1ddf2fc7a38d230f6539e3f7163c577416c5921 | [
{
"content": "A rectangle has a ratio of length to width of 5:2. If the diagonal of the rectangle is 13 units, determine the value of $k$ such that the area of the rectangle may be expressed as $kd^2$, where $d$ is the diagonal.",
"role": "user"
}
] | A rectangle has a ratio of length to width of 5:2. If the diagonal of the rectangle is 13 units, determine the value of $k$ such that the area of the rectangle may be expressed as $kd^2$, where $d$ is the diagonal. | "[\"\\\\frac{10}{29}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1512, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15507"}] | {
"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
} |
[
"515355128523864354"
] | [
"skywork"
] | 6d038420241ce84b428afababde72c8f4bf62fdb662dbf14e92d70297f2c86c1 | [
{
"content": "On each of the three axes, there is one rotating gear and one stationary pointer. The gears are connected sequentially. The first gear has 33 teeth, the second has 10, and the third has 7. On each tooth of the first gear, in clockwise order, one letter of the Russian alphabet is written in alphabe... | On each of the three axes, there is one rotating gear and one stationary pointer. The gears are connected sequentially. The first gear has 33 teeth, the second has 10, and the third has 7. On each tooth of the first gear, in clockwise order, one letter of the Russian alphabet is written in alphabetical order:
## А Б В... | "[\"515355128523864354\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1504, "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:15508"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"9\\sqrt{3}"
] | [
"skywork"
] | 3df9c816a21ad52e4e1482d53d415df6824c57268f0c15aab60ebda88480eab8 | [
{
"content": "Let $P_1, P_2, \\ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$ . Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon.",
"role": "user"
}
] | Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$ . Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon. | "[\"9\\\\sqrt{3}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1493, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "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:15509"}, {"dataset": "skywork", "extra_in... | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2}{n}"
] | [
"skywork"
] | 1f5756484c57523f38d39de225fa32b82d717584fa1e8b0f10ddb98978b62a28 | [
{
"content": "3. Given an integer $n \\geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \\cdots, z_{n}$, we have\n$$\n\\left|\\sum_{i=1}^{n} z_{i}\\right|+c \\sum_{1 \\leqslant i<j \\leqslant n}\\left|z_{i}-z_{j}\\right| \\geqslant \\sum_{i=1}^{n}\\left|z... | 3. Given an integer $n \geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \cdots, z_{n}$, we have
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leqslant i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right| .
$$
(Supplied by Zhang D... | "[\"\\\\frac{2}{n}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1508, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15510"}, {"dataset": "skywork", "extra_i... | {
"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
} |
[
"75600"
] | [
"skywork"
] | 49c368788ccd420777f4758f8017cb5be5e57640ea3751afce63886ac978508d | [
{
"content": "Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different).",
"role": "user"
}
] | Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). | "[\"75600\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1521, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "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:15511"}] | {
"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
} |
[
"101001_2"
] | [
"skywork"
] | 54333f792d34a49050e60ca0454c1e336c70d1342ba057131c4f03ef4a93bf03 | [
{
"content": "Add $110_2 + 101_2 + 1011_2 + 10011_2.$ Express your answer in base $2.$",
"role": "user"
}
] | Add $110_2 + 101_2 + 1011_2 + 10011_2.$ Express your answer in base $2.$ | "[\"101001_2\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1537, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15512"}] | {
"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
} |
[
"13.42"
] | [
"skywork"
] | f7d1c48d20084e34860fac451d551e68406f3282ed1ce5c7a17c51a34c99c909 | [
{
"content": "7. As shown in the right figure, the side length of the square is 2 meters, and the radius of the four circles is 1 meter, with the centers of the circles being the four vertices of the square. Question: What is the area covered by this square and the four circles in square meters?",
"role": "... | 7. As shown in the right figure, the side length of the square is 2 meters, and the radius of the four circles is 1 meter, with the centers of the circles being the four vertices of the square. Question: What is the area covered by this square and the four circles in square meters? | "[\"13.42\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1517, "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:15513"}] | {
"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
} |
[
"128"
] | [
"skywork"
] | a377eeb4de5082e365cb299233fe3b9891c97b3421e864d1a3a0c6016b150ef0 | [
{
"content": "Let $A=3^{7}+\\binom{7}{2}3^{5}+\\binom{7}{4}3^{3}+\\binom{7}{6}3$, $B=\\binom{7}{1}3^{6}+\\binom{7}{3}3^{4}+\\binom{7}{5}3^{2}+1$. Find $A-B$.",
"role": "user"
}
] | Let $A=3^{7}+\binom{7}{2}3^{5}+\binom{7}{4}3^{3}+\binom{7}{6}3$, $B=\binom{7}{1}3^{6}+\binom{7}{3}3^{4}+\binom{7}{5}3^{2}+1$. Find $A-B$. | "[\"128\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1525, "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:15514"}] | {
"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
} |
[
"44"
] | [
"skywork"
] | 829077f46ae49556758054c7531d73a6fa824485232945ca244c05ca08b49ed2 | [
{
"content": "Real numbers \\( x, y, z \\) satisfy \\( x \\geq y \\geq z \\geq 0 \\) and \\( 6x + 5y + 4z = 120 \\). Find the sum of the maximum and minimum values of \\( x + y + z \\).",
"role": "user"
}
] | Real numbers \( x, y, z \) satisfy \( x \geq y \geq z \geq 0 \) and \( 6x + 5y + 4z = 120 \). Find the sum of the maximum and minimum values of \( x + y + z \). | "[\"44\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1522, "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:15515"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-1 \\leqslant a \\leqslant \\frac{17}{8}"
] | [
"skywork"
] | bdec23ec54ed03dc4ad59d000e211c6138e71aa83dece13893c76dfb98ee4441 | [
{
"content": "Example 2 If the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points, then the range of the real number $a$ is $\\qquad$\n(1998, National High School Mathematics Competition)",
"role": "user"
}
] | Example 2 If the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points, then the range of the real number $a$ is $\qquad$
(1998, National High School Mathematics Competition) | "[\"-1 \\\\leqslant a \\\\leqslant \\\\frac{17}{8}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1527, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "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:15516"}] | {
"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
} |
[
"60718293"
] | [
"skywork"
] | a9b149fc934ce6eb145b06f85c1276c65090b37e82281521c94517183a802630 | [
{
"content": "Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.",
"role": "user"
}
] | Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7. | "[\"60718293\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1535, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15517"}] | {
"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_n = P"
] | [
"skywork"
] | 1d5da2871069150b1638c88ded1c37de391b13656053087d48bf45f7b5772f4f | [
{
"content": "Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \\in \\mathbb{N}$, such that for any prime $p$, $p > N$,\nthe $p$-th term of the progression is also prime.",
"role": "user"
}
] | Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$,
the $p$-th term of the progression is also prime. | "[\" a_n = P \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1499, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15518"}] | {
"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
} |
[
"\\left(\\frac{1}{2},\\frac{2}{3}\\right)"
] | [
"skywork"
] | 3c1ac74f794d11d1e5cebc5c056c01e710bc92bed591783bfab5161fac6750a9 | [
{
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.",
"role": "user"
}
] | Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | "[\"\\\\left(\\\\frac{1}{2},\\\\frac{2}{3}\\\\right)\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1540, "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:15519"}] | {
"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
} |
[
"607"
] | [
"skywork"
] | ca8fb47d74f6e90a9294bc1b52801f28806540d3da7eea4cf726a451124114ff | [
{
"content": "Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\\leq n\\leq2017$ does $5^n$ start with $1$?\n\n[i]2018 CCA Math Bonanza Tiebreaker Round #3[/i]",
"role": "user"
}
] | Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$?
[i]2018 CCA Math Bonanza Tiebreaker Round #3[/i] | "[\"607\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1536, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15520"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"50"
] | [
"skywork"
] | 0bf88d88f3e442cbe3bab078804742c4c89e3918f9adf3a9f75f19d7d72231e0 | [
{
"content": "Let $S=\\{1,2,\\ldots ,98\\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it... | Let $S=\{1,2,\ldots ,98\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other ... | "[\"50\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1524, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15521"}, {"dataset": "skywork", "extra_i... | {
"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
} |
[
"39"
] | [
"skywork"
] | ef196b475703078d11371a705a7d6d6bc0e962558a0de53b8a672d1fd3e2bd0c | [
{
"content": "The area of triangle \\(ABC\\) is 1. Points \\(B'\\), \\(C'\\), and \\(A'\\) are placed respectively on the rays \\(AB\\), \\(BC\\), and \\(CA\\) such that:\n\n\\[ BB' = 2 AB, \\quad CC' = 3 BC, \\quad AA' = 4 CA. \\]\n\nCalculate the area of triangle \\(A'B'C'\\).",
"role": "user"
}
] | The area of triangle \(ABC\) is 1. Points \(B'\), \(C'\), and \(A'\) are placed respectively on the rays \(AB\), \(BC\), and \(CA\) such that:
\[ BB' = 2 AB, \quad CC' = 3 BC, \quad AA' = 4 CA. \]
Calculate the area of triangle \(A'B'C'\). | "[\"39\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1514, "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:15522"}] | {
"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{\\sqrt{10}}{2}(n+1)\\sqrt{M}"
] | [
"skywork"
] | 0c3dd727332466a118203fa84961969d0d646c786184ba507f943a6348effa19 | [
{
"content": "Five. (Full marks 20 points) Given a positive integer $n$ and a positive number $M$, for all arithmetic sequences $a_{1}, a_{2}, a_{3}, \\cdots$ satisfying the condition $a_{1}^{2}+a_{n+1}^{2} \\leqslant M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\\cdots+a_{2 n+1}$.",
"role": "user"
}
... | Five. (Full marks 20 points) Given a positive integer $n$ and a positive number $M$, for all arithmetic sequences $a_{1}, a_{2}, a_{3}, \cdots$ satisfying the condition $a_{1}^{2}+a_{n+1}^{2} \leqslant M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$. | "[\"\\\\frac{\\\\sqrt{10}}{2}(n+1)\\\\sqrt{M}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1528, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "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:15523"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"9.6"
] | [
"skywork"
] | 84fdfd5a6838758f75b1ef7e419d74298f33def58bd05bf1aa9a5d9b45862f3d | [
{
"content": "The height of trapezoid $A B C D$ is 7, the bases $A D$ and $B C$ are 8 and 6 respectively. A line through point $E$, lying on side $C D$, intersects diagonal $A C$ at point $O$ in the ratio $A O: O C=3: 2$. Find the area of triangle $O E C$.\n\n#",
"role": "user"
}
] | The height of trapezoid $A B C D$ is 7, the bases $A D$ and $B C$ are 8 and 6 respectively. A line through point $E$, lying on side $C D$, intersects diagonal $A C$ at point $O$ in the ratio $A O: O C=3: 2$. Find the area of triangle $O E C$.
# | "[\"9.6\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1531, "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:15524"}] | {
"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
} |
[
"11"
] | [
"skywork"
] | 2a3e815d6b91cf47d388ad8413ef97ad9f7988a1d64094deaa98c902462dd4ef | [
{
"content": "Example 31 (2004 China Mathematical Olympiad) $M$ is a set of $n$ points in the plane, satisfying:\n(1) $M$ contains 7 points which are the 7 vertices of a convex heptagon;\n(2) For any 5 points in $M$, if these 5 points are the 5 vertices of a convex pentagon, then this convex pentagon contains a... | Example 31 (2004 China Mathematical Olympiad) $M$ is a set of $n$ points in the plane, satisfying:
(1) $M$ contains 7 points which are the 7 vertices of a convex heptagon;
(2) For any 5 points in $M$, if these 5 points are the 5 vertices of a convex pentagon, then this convex pentagon contains at least one point of $M$... | "[\"11\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1520, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "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:15525"}] | {
"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
} |
[
"203"
] | [
"skywork"
] | d843812e2b47a5b5a8f237714f1eede1b7e13b106d29b2405cbece15eaf48564 | [
{
"content": "Farmer James has some strange animals. His hens have 2 heads and 8 legs, his peacocks have 3 heads and 9 legs, and his zombie hens have 6 heads and 12 legs. Farmer James counts 800 heads and 2018 legs on his farm. What is the number of animals that Farmer James has on his farm?",
"role": "user... | Farmer James has some strange animals. His hens have 2 heads and 8 legs, his peacocks have 3 heads and 9 legs, and his zombie hens have 6 heads and 12 legs. Farmer James counts 800 heads and 2018 legs on his farm. What is the number of animals that Farmer James has on his farm? | "[\"203\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1530, "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:15526"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"skywork"
] | 18ab2bd625bcd36fbcde104271e26ccda2732da548b56ed63c2f615c09dc7e12 | [
{
"content": "Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying\n\\begin{align*}\n 2a_1+a_2+a_3+a_4+a_5 &= 1 + \\tfrac{1}{8}a_4 \\\\\n 2a_2+a_3+a_4+a_5 &= 2 + \\tfrac{1}{4}a_3 \\\\\n 2a_3+a_4+a_5 &= 4 + \\tfrac{1}{2}a_2 \\\\\n 2a_4+a_5 &= 6 + a_1\n\\end{align*}\nComput... | Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying
\begin{align*}
2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\
2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\
2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\
2a_4+a_5 &= 6 + a_1
\end{align*}
Compute $a_1+a_2+a_3+a_4+a_5$.
[i]Proposed by... | "[\"2\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1538, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "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:15527"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{\\sum_{i=1}^{n}(\\sqrt{i}-\\sqrt{i-1})^{2}}"
] | [
"skywork"
] | 71350de79cf170f57421f2e852ff231c42d4d7be75d459ae77bc5a1449c2dbc4 | [
{
"content": "Example 11 Let $x_{i} \\geqslant 0, i=1,2, \\cdots, n$, satisfy\n$$\\sum_{i=1}^{n} x_{i}^{2}+2 \\sum_{1 \\leq j<k \\leqslant n} \\sqrt{\\frac{j}{k}} x_{j} x_{k}=1 .$$\n\nFind the minimum and maximum values of $x_{1}+x_{2}+\\cdots+x_{n}$.",
"role": "user"
}
] | Example 11 Let $x_{i} \geqslant 0, i=1,2, \cdots, n$, satisfy
$$\sum_{i=1}^{n} x_{i}^{2}+2 \sum_{1 \leq j<k \leqslant n} \sqrt{\frac{j}{k}} x_{j} x_{k}=1 .$$
Find the minimum and maximum values of $x_{1}+x_{2}+\cdots+x_{n}$. | "[\"\\\\sqrt{\\\\sum_{i=1}^{n}(\\\\sqrt{i}-\\\\sqrt{i-1})^{2}}\"]" | train-math-numinamath1.5_inequalities | [{"dataset": "skywork", "extra_info": {"index": 1532, "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:15528"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"3\\sqrt{3}"
] | [
"skywork"
] | 4266a11aa51564407f439036c985443b1dcbeaf1283f7e363e06cca0eed4e542 | [
{
"content": "Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\\sin t)$ .\n \nFive minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim.... | Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way t... | "[\"3\\\\sqrt{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1496, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "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:15529"}] | {
"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
} |
[
"73"
] | [
"skywork"
] | 3db1f9d0de961d5d902f2f02059c915a65440de802f0398dda51a378c9c259c5 | [
{
"content": "23 Find the number of integer pairs $(a, b)$ that satisfy the following conditions: $0 \\leqslant a, b \\leqslant 36$, and $a^{2}+b^{2}=$ $0(\\bmod 37)$.",
"role": "user"
}
] | 23 Find the number of integer pairs $(a, b)$ that satisfy the following conditions: $0 \leqslant a, b \leqslant 36$, and $a^{2}+b^{2}=$ $0(\bmod 37)$. | "[\"73\"]" | train-math-numinamath1.5_number_theory | [{"dataset": "skywork", "extra_info": {"index": 1542, "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:15530"}] | {
"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
} |
[
"\\pi"
] | [
"skywork"
] | d01268bac7c6df54fa9c92c4df0f850bbedbe981d35b8caa5c82c5365d315e06 | [
{
"content": "Let the function \n$$\nf(x) = A \\sin(\\omega x + \\varphi) \\quad (A>0, \\omega>0).\n$$\n\nIf \\( f(x) \\) is monotonic on the interval \\(\\left[\\frac{\\pi}{6}, \\frac{\\pi}{2}\\right]\\) and \n$$\nf\\left(\\frac{\\pi}{2}\\right) = f\\left(\\frac{2\\pi}{3}\\right) = -f\\left(\\frac{\\pi}{6}\\ri... | Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$
If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$
then the smallest positive period of \( f(x) \... | "[\"\\\\pi\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1523, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15531"}] | {
"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\\mathrm{~}"
] | [
"skywork"
] | 3998e06d1d9808f50ee4dae6be363eaffbe2afc35403060c58d6865c0e59ee76 | [
{
"content": "1. The lengths of two sides of a triangle are $5 \\mathrm{~cm}$ and $2 \\mathrm{~cm}$. The length of the third side in $\\mathrm{cm}$ is an odd integer. What is the length of the third side?\nA $1 \\mathrm{~cm}$\nB $3 \\mathrm{~cm}$\nC $5 \\mathrm{~cm}$\nD $7 \\mathrm{~cm}$\nE $9 \\mathrm{~cm}$",
... | 1. The lengths of two sides of a triangle are $5 \mathrm{~cm}$ and $2 \mathrm{~cm}$. The length of the third side in $\mathrm{cm}$ is an odd integer. What is the length of the third side?
A $1 \mathrm{~cm}$
B $3 \mathrm{~cm}$
C $5 \mathrm{~cm}$
D $7 \mathrm{~cm}$
E $9 \mathrm{~cm}$ | "[\"5\\\\mathrm{~}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1534, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 10, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15532"}] | {
"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{92}{61}"
] | [
"skywork"
] | d8379bc88614a7194de2e638c9ffa551d86a10053e06593d5df44b213f5cd915 | [
{
"content": "Two circles are said to be [i]orthogonal[/i] if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\\omega_1$ and $\\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\\omega_3$ with r... | Two circles are said to be [i]orthogonal[/i] if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_1$ and $\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\omega_3$ with radius $2\sqrt2$ passes th... | "[\"\\\\frac{92}{61}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1518, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15533"}] | {
"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 + \\sqrt{35}"
] | [
"skywork"
] | 061d6c95ea4adcc90b2a41d1dece1d06e34fcb416cfdff80dfc835f00579660c | [
{
"content": "Let right $\\triangle ABC$ have $AC = 3$, $BC = 4$, and right angle at $C$. Let $D$ be the projection from $C$ to $\\overline{AB}$. Let $\\omega$ be a circle with center $D$ and radius $\\overline{CD}$, and let $E$ be a variable point on the circumference of $\\omega$. Let $F$ be the reflection of... | Let right $\triangle ABC$ have $AC = 3$, $BC = 4$, and right angle at $C$. Let $D$ be the projection from $C$ to $\overline{AB}$. Let $\omega$ be a circle with center $D$ and radius $\overline{CD}$, and let $E$ be a variable point on the circumference of $\omega$. Let $F$ be the reflection of $E$ over point $D$, and le... | "[\" 12 + \\\\sqrt{35} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1526, "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:15534"}] | {
"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{131\\sqrt{3}}{42}"
] | [
"skywork"
] | a73e2bca659a7c7a8416b639d508307d50f0077b53038903e6ec4ebd1db1a6a1 | [
{
"content": "In triangle $XYZ$, $XY = 5$, $XZ = 7$, and $YZ = 8$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$ respectively. Find $GP + GQ + GR$.",
"role": "user"
}
] | In triangle $XYZ$, $XY = 5$, $XZ = 7$, and $YZ = 8$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$ respectively. Find $GP + GQ + GR$. | "[\"\\\\frac{131\\\\sqrt{3}}{42}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1539, "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:15535"}] | {
"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=y=z=2"
] | [
"skywork"
] | 7214944b53e1f14310a7f0dd04f311f088788e07259e1b8f36926817788c4fea | [
{
"content": "24 Find all positive integers $x, y, z$, such that $3^{x}+4^{y}=5^{z}$.",
"role": "user"
}
] | 24 Find all positive integers $x, y, z$, such that $3^{x}+4^{y}=5^{z}$. | "[\"x=y=z=2\"]" | train-math-numinamath1.5_number_theory | [{"dataset": "skywork", "extra_info": {"index": 1551, "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:15536"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | 63a5de23f0f2f44c3e2968256e1830e24f5a898b57cc54eb1d89961b84963bcd | [
{
"content": "Let $ f(x)\\equal{}\\frac{1}{\\sin x\\sqrt{1\\minus{}\\cos x}}\\ (0<x<\\pi)$.\r\n\r\n(1) Find the local minimum value of $ f(x)$.\r\n\r\n(2) Evaluate $ \\int_{\\frac{\\pi}{2}}^{\\frac{2\\pi}{3}} f(x)\\ dx$.",
"role": "user"
}
] | Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$.
(1) Find the local minimum value of $ f(x)$.
(2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$. | "[\"1\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1529, "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:15537"}] | {
"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{5}"
] | [
"skywork"
] | 17bdd29e810222182bd25582a7178e2d5d89495c530ec6c645c782e5c08063b8 | [
{
"content": "In parallelogram $ABCD$, $AB = 10$, and $AB = 2BC$. Let $M$ be the midpoint of $CD$, and suppose that $BM = 2AM$. Compute $AM$.",
"role": "user"
}
] | In parallelogram $ABCD$, $AB = 10$, and $AB = 2BC$. Let $M$ be the midpoint of $CD$, and suppose that $BM = 2AM$. Compute $AM$. | "[\"2\\\\sqrt{5}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1550, "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:15538"}, {"dataset": "skywork", "extra_inf... | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{\\sqrt{5}-1}{2} d"
] | [
"skywork"
] | b27e166af0a6f016cc7252fc13780a96c5488749f7b73dace8f8dc6a43bceed0 | [
{
"content": "Four, (25 points) As shown in the figure,\n$3, AB$ is the diameter of $\\odot O$, $AB=d$, a tangent line to $\\odot O$ is drawn through point $A$ and a point $C$ is taken on it such that $AC = AB$, connect $OC$ to intersect $\\odot O$ at point $D$, the extension of $BD$ intersects $AC$ at $E$. Fin... | Four, (25 points) As shown in the figure,
$3, AB$ is the diameter of $\odot O$, $AB=d$, a tangent line to $\odot O$ is drawn through point $A$ and a point $C$ is taken on it such that $AC = AB$, connect $OC$ to intersect $\odot O$ at point $D$, the extension of $BD$ intersects $AC$ at $E$. Find the length of $AE$.
保留源... | "[\"\\\\frac{\\\\sqrt{5}-1}{2} d\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1549, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15539"}] | {
"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{\\sqrt{10}}{4}"
] | [
"skywork"
] | 101750e16f9343bcf04ab54e36a5836750162b4c0e99b9a1570bb1bdb4ea4d21 | [
{
"content": "7. Let $O$ be the circumcenter of $\\triangle A B C$. If $\\overrightarrow{A O}=\\overrightarrow{A B}+2 \\overrightarrow{A C}$, then the value of $\\sin \\angle B A C$ is $\\qquad$",
"role": "user"
}
] | 7. Let $O$ be the circumcenter of $\triangle A B C$. If $\overrightarrow{A O}=\overrightarrow{A B}+2 \overrightarrow{A C}$, then the value of $\sin \angle B A C$ is $\qquad$ | "[\"\\\\frac{\\\\sqrt{10}}{4}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1543, "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:15540"}] | {
"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
} |
[
"7938"
] | [
"skywork"
] | f2725261542fa90bb52ac80b4669777ee7a8e15084344f813c7315eece5c31ff | [
{
"content": "23. Let $a=\\frac{\\sqrt{5}+\\sqrt{3}}{\\sqrt{5}-\\sqrt{3}}$ and $b=\\frac{\\sqrt{5}-\\sqrt{3}}{\\sqrt{5}+\\sqrt{3}}$. Find the value of $a^{4}+b^{4}+(a+b)^{4}$.",
"role": "user"
}
] | 23. Let $a=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $b=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$. Find the value of $a^{4}+b^{4}+(a+b)^{4}$. | "[\"7938\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1560, "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:15541"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"8"
] | [
"skywork"
] | bf7f9b3b591bc1d8278b82dcd4ef77f2f60da29b65bddd1cb58a49037d9e5a84 | [
{
"content": "11. Xiao Ming brought 24 handmade souvenirs to sell at the London Olympics. In the morning, each souvenir was sold for 7 pounds, and the number of souvenirs sold was less than half of the total. In the afternoon, he discounted the price of each souvenir, and the discounted price was still an integ... | 11. Xiao Ming brought 24 handmade souvenirs to sell at the London Olympics. In the morning, each souvenir was sold for 7 pounds, and the number of souvenirs sold was less than half of the total. In the afternoon, he discounted the price of each souvenir, and the discounted price was still an integer. In the afternoon, ... | "[\"8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1544, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15542"}] | {
"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
} |
[
"1482"
] | [
"skywork"
] | f92413d8d49981ad91fc9808caf8b9f71f7276da4b6a47768e73901f041bdda7 | [
{
"content": "4. Given positive integers $a, b, c$ are all distinct, and satisfy $a b c=72^{3}$. Then the number of ordered positive integer triples $(a, b, c)$ is $\\qquad$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and output the translation result dir... | 4. Given positive integers $a, b, c$ are all distinct, and satisfy $a b c=72^{3}$. Then the number of ordered positive integer triples $(a, b, c)$ is $\qquad$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | "[\"1482\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1566, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15543"}] | {
"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
} |
[
"F_{n+1}"
] | [
"skywork"
] | 8230ce99a76a1a926e99ced5102402f0befcb27af02f6f84c2f74c15b25b2660 | [
{
"content": "For any positive integer n, evaluate $$\\sum_{i=0}^{\\lfloor \\frac{n+1}{2} \\rfloor}\n {n-i+1 \\choose i}$$\n, where $\\lfloor n \\rfloor$ is the greatest integer less than or equal to $n$ .",
"role": "user"
}
] | For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor}
{n-i+1 \choose i}$$
, where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ . | "[\" F_{n+1} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1562, "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:15544"}] | {
"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
} |
[
"72\\mathrm{}"
] | [
"skywork"
] | 4e1595b549c4b242094577df4427b6fd05ae54ec0bd76ed9c7d2b89afd54ece8 | [
{
"content": "4. (15 points) Two heaters are connected sequentially to the same DC power source. The water in the pot boiled after $t_{1}=120$ s from the first heater. The same water, taken at the same initial temperature, boiled after $t_{2}=180 \\mathrm{s}$ from the second heater. How long would it take for t... | 4. (15 points) Two heaters are connected sequentially to the same DC power source. The water in the pot boiled after $t_{1}=120$ s from the first heater. The same water, taken at the same initial temperature, boiled after $t_{2}=180 \mathrm{s}$ from the second heater. How long would it take for the water to boil if the... | "[\"72\\\\mathrm{}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1541, "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:15545"}] | {
"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
} |
[
"616"
] | [
"skywork"
] | dc5d8e50e10eb879e0e5d474f01029b99022f87168717823c29335d27310e1b4 | [
{
"content": "If $\\sqrt{n+9}=25$, then $n$ equals\n(A) 256\n(B) -4\n(C) 484\n(D) 616\n(E) 16",
"role": "user"
}
] | If $\sqrt{n+9}=25$, then $n$ equals
(A) 256
(B) -4
(C) 484
(D) 616
(E) 16 | "[\"616\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1577, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15546"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2\\sqrt{3}}{3}"
] | [
"skywork"
] | 9ca72becadbc23608c11b85c314096fefa2dbbed561ed0d2c24929c1269701c5 | [
{
"content": "5. Given four points in space $A, B, C, D$ satisfying $A B \\perp A C, A B \\perp A D, A C \\perp A D$, and $A B=A C=A D=1, Q$ is a moving point on the circumscribed sphere of the tetrahedron $A-B C D$, then the maximum distance from point $Q$ to the plane $B C D$ is",
"role": "user"
}
] | 5. Given four points in space $A, B, C, D$ satisfying $A B \perp A C, A B \perp A D, A C \perp A D$, and $A B=A C=A D=1, Q$ is a moving point on the circumscribed sphere of the tetrahedron $A-B C D$, then the maximum distance from point $Q$ to the plane $B C D$ is | "[\"\\\\frac{2\\\\sqrt{3}}{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1545, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15547"}] | {
"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]{2}"
] | [
"skywork"
] | ed62fde88ea60814a04ea98cf497f566fdb85fd0bd93e9b2aa8b98998aa1f502 | [
{
"content": "Consider a cylinder and a cone with a common base such that the volume of the\r\npart of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.",
"role": "user"
}
] | Consider a cylinder and a cone with a common base such that the volume of the
part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder. | "[\" \\\\sqrt[3]{2} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1555, "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:15548"}] | {
"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
} |
[
"15"
] | [
"skywork"
] | 86d2d6253304bb3d357f22c921b4ad705f189804b08ec93352f1c9b6e2551eb2 | [
{
"content": "25. Given that the internal angles of $\\triangle A_{0} B_{0} C_{0}$ are $59.999^{\\circ}$, $60^{\\circ}$, and $60.001^{\\circ}$. Let $A_{n}\\left(n \\in \\mathbf{Z}_{+}\\right)$ be the projection of point $A_{n-1}$ on side $B_{n-1} C_{n-1}$, $B_{n}$ and $C_{n}$ be the projections of points $B_{n-... | 25. Given that the internal angles of $\triangle A_{0} B_{0} C_{0}$ are $59.999^{\circ}$, $60^{\circ}$, and $60.001^{\circ}$. Let $A_{n}\left(n \in \mathbf{Z}_{+}\right)$ be the projection of point $A_{n-1}$ on side $B_{n-1} C_{n-1}$, $B_{n}$ and $C_{n}$ be the projections of points $B_{n-1}$ and $C_{n-1}$ on sides $C_... | "[\"15\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1546, "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:15549"}] | {
"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{\\sqrt{5} + 1}{2}"
] | [
"skywork"
] | 495be87f3e830498f608c58de6e1de4c2a1fff6446b36dd44e46e40bbd56cca3 | [
{
"content": "Find the maximum value of $\\lambda ,$ such that for $\\forall x,y\\in\\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\\lambda y^2\\leqslant 1.$",
"role": "user"
}
] | Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ | "[\"\\\\frac{\\\\sqrt{5} + 1}{2}\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1567, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "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:15550"}, {"dataset": "skywork", "extra_in... | {
"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
} |
[
"17"
] | [
"skywork"
] | 8c66446437f57d4283c0badd54ee6b592d7bca74a0a4160d55f6655b08c3b724 | [
{
"content": "3. Let $m$ be the largest integer that cannot be expressed as the sum of three distinct composite numbers. Then $m=$ $\\qquad$\n\nSet $m$ is the largest integer that cannot be represented as the sum of three distinct composite numbers. Then $m=$ $\\qquad$",
"role": "user"
}
] | 3. Let $m$ be the largest integer that cannot be expressed as the sum of three distinct composite numbers. Then $m=$ $\qquad$
Set $m$ is the largest integer that cannot be represented as the sum of three distinct composite numbers. Then $m=$ $\qquad$ | "[\"17\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1556, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15551"}] | {
"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
} |
[
"1992"
] | [
"skywork"
] | a86b6940218b6567c97735a0c3070406704b71d4543f61b51c574a5c1f201eaf | [
{
"content": "Given \\( x_{1}, x_{2}, \\cdots, x_{1993} \\) satisfy:\n\\[\n\\left|x_{1}-x_{2}\\right|+\\left|x_{2}-x_{3}\\right|+\\cdots+\\left|x_{1992}-x_{1993}\\right|=1993,\n\\]\nand\n\\[\ny_{k}=\\frac{x_{1}+x_{2}+\\cdots+x_{k}}{k} \\quad (k=1,2,\\cdots,1993),\n\\]\nwhat is the maximum possible value of \\( ... | Given \( x_{1}, x_{2}, \cdots, x_{1993} \) satisfy:
\[
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993,
\]
and
\[
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k} \quad (k=1,2,\cdots,1993),
\]
what is the maximum possible value of \( \left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|... | "[\"1992\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1548, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "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:15552"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{2\\sqrt{21}+3}{5}"
] | [
"skywork"
] | bd341939a2639a2dda93ae595a06df323313aab7c1b27245a56a1efe8fd289fc | [
{
"content": "9. Given $\\triangle A B C$ satisfies $A B=1, A C=2, \\cos B+\\sin C=1$, find the length of side $B C$.\n\n---\n\nThe translation is provided as requested, maintaining the original format and line breaks.",
"role": "user"
}
] | 9. Given $\triangle A B C$ satisfies $A B=1, A C=2, \cos B+\sin C=1$, find the length of side $B C$.
---
The translation is provided as requested, maintaining the original format and line breaks. | "[\"\\\\frac{2\\\\sqrt{21}+3}{5}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1565, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15553"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"4"
] | [
"skywork"
] | 70ef356778976148d48addb05c7fa3caf73ae0f27cb6dc99d8f9704616540b0f | [
{
"content": "2B. Let $a$ and $b$ be rational numbers such that $a>1, b>0$ and $a b=a^{b}$ and $\\frac{a}{b}=a^{3 b}$. Determine the values of $a$?",
"role": "user"
}
] | 2B. Let $a$ and $b$ be rational numbers such that $a>1, b>0$ and $a b=a^{b}$ and $\frac{a}{b}=a^{3 b}$. Determine the values of $a$? | "[\"4\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1553, "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:15554"}] | {
"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
} |
[
"46"
] | [
"skywork"
] | 9f0c5090e78fcf7c6867828d4c4ad1e8e4fc846873e00976042bd1d00c8a182c | [
{
"content": "24. There are two ponds at the bottom of Gabrielle's garden, each containing frogs and toads. In one pond the ratio of frogs to toads is $3: 4$. In the other pond the ratio of frogs to toads is $5: 6$. Suppose there are 36 frogs in total. What then would be the largest possible total number of toa... | 24. There are two ponds at the bottom of Gabrielle's garden, each containing frogs and toads. In one pond the ratio of frogs to toads is $3: 4$. In the other pond the ratio of frogs to toads is $5: 6$. Suppose there are 36 frogs in total. What then would be the largest possible total number of toads in the ponds?
A 48
... | "[\"46\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1569, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15555"}] | {
"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
} |
[
"PZ"
] | [
"skywork"
] | 399b80ac058836d664ab71072c164428cfb8ea30266ad01339f24c97ab9731ed | [
{
"content": "Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$.",
"role": "user"
}
] | Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$. | "[\" PZ \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1552, "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:15556"}] | {
"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
} |
[
"48"
] | [
"skywork"
] | 99774dfd35e06f2cdbe0afd651ce7aa79dee52774db84ee6d5acc0029caa8c26 | [
{
"content": "The Gnollish language now consists of 4 words: `splargh`, `glumph`, `amr`, and `drung`. In a sentence, `splargh` cannot come directly before `glumph`, and `amr` cannot come directly after `drung`. How many valid 3-word sentences are there in Gnollish?",
"role": "user"
}
] | The Gnollish language now consists of 4 words: `splargh`, `glumph`, `amr`, and `drung`. In a sentence, `splargh` cannot come directly before `glumph`, and `amr` cannot come directly after `drung`. How many valid 3-word sentences are there in Gnollish? | "[\"48\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 1583, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15557"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"64"
] | [
"skywork"
] | 193d3394acdd29c7557c9e5d0a08670994c9010fe0696638d8892f584213d9a1 | [
{
"content": "SI. 4 If $c=\\log _{4} d$, find $d$.",
"role": "user"
}
] | SI. 4 If $c=\log _{4} d$, find $d$. | "[\"64\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1561, "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:15558"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | f06bd8374f2c2c00ad0d737a3c60b152f30a84bc083bf3ec8b047a5e916f9741 | [
{
"content": "156. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Points $M$ and $N$ are taken on segments $A A_{1}$ and $B C_{1}$ such that the line $M N$ intersects the line $B_{1} D$. Find\n\n$$\n\\frac{\\left|B C_{1}\\right|}{|B N|}-\\frac{|A M|}{\\left|A A_{1}\\right|}\n$$",
"role": "user"
}
] | 156. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Points $M$ and $N$ are taken on segments $A A_{1}$ and $B C_{1}$ such that the line $M N$ intersects the line $B_{1} D$. Find
$$
\frac{\left|B C_{1}\right|}{|B N|}-\frac{|A M|}{\left|A A_{1}\right|}
$$ | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1564, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "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:15559"}] | {
"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
} |
[
"531"
] | [
"skywork"
] | 09be2fb232710e30d283fea8b430fb32d38481f5d112eaa123e03b47a557125e | [
{
"content": "For a positive integer \\( n \\), let \\( s(n) \\) denote the sum of its digits, and let \\( p(n) \\) denote the product of its digits. If the equation \\( s(n) + p(n) = n \\) holds true, then \\( n \\) is called a coincidence number. What is the sum of all coincidence numbers?",
"role": "user... | For a positive integer \( n \), let \( s(n) \) denote the sum of its digits, and let \( p(n) \) denote the product of its digits. If the equation \( s(n) + p(n) = n \) holds true, then \( n \) is called a coincidence number. What is the sum of all coincidence numbers? | "[\"531\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1557, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15560"}] | {
"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^{2016}-1"
] | [
"skywork"
] | f30c295b36e3a66986f10516295286972c5df4c7127b718c1e813588fa591898 | [
{
"content": "3. A non-empty finite set of numbers is called a trivial set if the sum of the squares of all its elements is odd.\n\nLet the set $A=\\{1,2, \\cdots, 2017\\}$. Then the number of trivial sets among all proper subsets of $A$ is $\\qquad$ (powers of numbers are allowed in the answer).",
"role": ... | 3. A non-empty finite set of numbers is called a trivial set if the sum of the squares of all its elements is odd.
Let the set $A=\{1,2, \cdots, 2017\}$. Then the number of trivial sets among all proper subsets of $A$ is $\qquad$ (powers of numbers are allowed in the answer). | "[\"2^{2016}-1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1571, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:15561"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{2}"
] | [
"skywork"
] | 2b5f2a93b56389fd123ba2c3c43af62f6a91fef8e8eddb76a9d37678c36cad97 | [
{
"content": "6.2. Mathematicians Andrey, Boris, and Viktor were solving problems from an olympiad. First, Andrey solved several problems, then Boris solved a third of the remaining problems. After this, a third of the problems remained unsolved, which Viktor completed. What part of all the problems did Andrey ... | 6.2. Mathematicians Andrey, Boris, and Viktor were solving problems from an olympiad. First, Andrey solved several problems, then Boris solved a third of the remaining problems. After this, a third of the problems remained unsolved, which Viktor completed. What part of all the problems did Andrey solve? | "[\"\\\\frac{1}{2}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1558, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15562"}] | {
"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
} |
[
"70"
] | [
"skywork"
] | 76905616333b895719adcbd3aeb03c936400ced5ac623137e7cb674465d368ab | [
{
"content": "In the hexagon $ABCDEF$ with vertices $A, B, C, D, E, F$, all internal triangles dividing the hexagon are similar to isosceles triangle $ABC$, where $AB = AC$. Among these triangles, there are $10$ smallest triangles each with area $2$, and the area of triangle $ABC$ is $80$. Determine the area of... | In the hexagon $ABCDEF$ with vertices $A, B, C, D, E, F$, all internal triangles dividing the hexagon are similar to isosceles triangle $ABC$, where $AB = AC$. Among these triangles, there are $10$ smallest triangles each with area $2$, and the area of triangle $ABC$ is $80$. Determine the area of the quadrilateral $DB... | "[\"70\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1574, "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:15563"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0"
] | [
"skywork"
] | b693313724c758af8e4cd3c40dc2dc6016494995c88b65eb38ed02c87134520d | [
{
"content": "What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions?",
"role": "user"
}
] | What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | "[\"0\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1576, "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:15564"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"5 \\sqrt{11}"
] | [
"skywork"
] | e9c39b5be48c3c2aea345db9b5612308cb90e2d35e04225f7dd0cce8a593217e | [
{
"content": "3. Let $a, b \\in \\mathbf{R}$, and $a+b=1$. Then\n$$\nf(a, b)=3 \\sqrt{1+2 a^{2}}+2 \\sqrt{40+9 b^{2}}\n$$\n\nthe minimum value of $f(a, b)$ is $\\qquad$",
"role": "user"
}
] | 3. Let $a, b \in \mathbf{R}$, and $a+b=1$. Then
$$
f(a, b)=3 \sqrt{1+2 a^{2}}+2 \sqrt{40+9 b^{2}}
$$
the minimum value of $f(a, b)$ is $\qquad$ | "[\"5 \\\\sqrt{11}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1559, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "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:15565"}] | {
"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
} |
[
"991 * 997 * 1009"
] | [
"skywork"
] | 5d8295ef41c12dd19578a3de465148c48d298cb2421e324675a7225dfb6b03da | [
{
"content": "Factorize the number \\( 989 \\cdot 1001 \\cdot 1007 + 320 \\) into prime factors.",
"role": "user"
}
] | Factorize the number \( 989 \cdot 1001 \cdot 1007 + 320 \) into prime factors. | "[\"991 * 997 * 1009\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1585, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15566"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{12}"
] | [
"skywork"
] | 254b36db5422e5119b35dddbe1e21440887912a39793dcde91f71ef874e002c3 | [
{
"content": "A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?",
"role": "user"
}
] | A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | "[\"\\\\frac{1}{12}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1570, "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:15567"}] | {
"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
} |
[
"7"
] | [
"skywork"
] | 52056d196f84649bc733cba871d68d87f1be8cc52427f22400c07317f69e3cc7 | [
{
"content": "Example 3 Find the largest integer $n$, such that all non-zero solutions of the equation $(z+1)^{n}=z^{n}+1$ lie on the unit circle.",
"role": "user"
}
] | Example 3 Find the largest integer $n$, such that all non-zero solutions of the equation $(z+1)^{n}=z^{n}+1$ lie on the unit circle. | "[\"7\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1588, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15568"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0"
] | [
"skywork"
] | 41f3cf97bb3c46eda5e550df04381412da1182e559e08865b24a70a877b5a538 | [
{
"content": "9. Let $\\left\\{a_{n}\\right\\}$ be a sequence of positive terms in a geometric progression, and let $S_{n}=\\lg a_{1}+\\lg a_{2}+\\cdots+\\lg a_{n}, \\forall n \\in \\mathbf{N}^{*}$. If there exist distinct positive integers $m 、 n$, such that $S_{n}=S_{m}$, find $S_{m+n}$.",
"role": "user"
... | 9. Let $\left\{a_{n}\right\}$ be a sequence of positive terms in a geometric progression, and let $S_{n}=\lg a_{1}+\lg a_{2}+\cdots+\lg a_{n}, \forall n \in \mathbf{N}^{*}$. If there exist distinct positive integers $m 、 n$, such that $S_{n}=S_{m}$, find $S_{m+n}$. | "[\"0\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1573, "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:15569"}] | {
"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
} |
[
"11"
] | [
"skywork"
] | 3654237451880d97e6d5a22781865aa6d50ab7cb495235e6fbe2039d06efca8f | [
{
"content": "How many integers at minimum must be selected from the set $\\{1, 2, \\ldots, 20\\}$ to ensure that this selection includes two integers \\(a\\) and \\(b\\) such that \\(a - b = 2\\)?",
"role": "user"
}
] | How many integers at minimum must be selected from the set $\{1, 2, \ldots, 20\}$ to ensure that this selection includes two integers \(a\) and \(b\) such that \(a - b = 2\)? | "[\"11\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1572, "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:15570"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"20"
] | [
"skywork"
] | 59aa2c8b6a251405f42cb59bc3f1b69ef3aba530cdc7c26448d751f8d28dd864 | [
{
"content": "A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria?",
"role": "user"
}
] | A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria? | "[\"20\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1598, "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:15571"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"16"
] | [
"skywork"
] | 2b35f4eaf7bc1049c0dfeb999a681807c78f4d1bb36e5099e4b64b724bad35c9 | [
{
"content": "At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many p... | At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many players are taking chem... | "[\"16\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1591, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15572"}] | {
"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
} |
[
"n"
] | [
"skywork"
] | 8aeeeedf5fa0c9c8aaa58c99109553b62a2e55f2a851d2b48d4ad824576b9f23 | [
{
"content": "3. In an $n \\times n$ grid $C$, the numbers $1,2, \\ldots, n^{2}$ are written, with one number in each cell. If two cells share a common edge, they are called \"adjacent.\" Now we calculate the difference between the numbers in any two adjacent cells, and denote the maximum of all these differenc... | 3. In an $n \times n$ grid $C$, the numbers $1,2, \ldots, n^{2}$ are written, with one number in each cell. If two cells share a common edge, they are called "adjacent." Now we calculate the difference between the numbers in any two adjacent cells, and denote the maximum of all these differences as $g$. Find the minimu... | "[\"n\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1579, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 4, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15573"}] | {
"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"
] | 74124166924bb9a9c181dbccd7c632dddcfba4f9a57591af0e7c437974aacf9f | [
{
"content": "12. Given the ellipse $\\frac{x^{2}}{6}+\\frac{y^{2}}{2}=1$ with its right focus at $F$, the line $y=k(x-2)$ through $F$ intersects the ellipse at points $P$ and $Q$ $(k \\neq 0)$. If the midpoint of $P Q$ is $N$, and $O$ is the origin, the line $O N$ intersects the line $x=3$ at $M$.\n(1) Find th... | 12. Given the ellipse $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$ with its right focus at $F$, the line $y=k(x-2)$ through $F$ intersects the ellipse at points $P$ and $Q$ $(k \neq 0)$. If the midpoint of $P Q$ is $N$, and $O$ is the origin, the line $O N$ intersects the line $x=3$ at $M$.
(1) Find the size of $\angle M F Q$;
... | "[\"\\\\sqrt{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1575, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15574"}] | {
"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
} |
[
"65"
] | [
"skywork"
] | 6dc41f583225edf5760ea5765bcb0b274f1670af99cfe1bc32570ee9a9c84e51 | [
{
"content": "10. In $\\triangle A B C$, $A B=\\sqrt{30}, A C=\\sqrt{6}, B C$ $=\\sqrt{15}$, there is a point $D$ such that $A D$ bisects $B C$ and $\\angle A D B$ is a right angle, the ratio $\\frac{S_{\\triangle A D B}}{S_{\\triangle A B C}}$ can be written as $\\frac{m}{n}$. ($m, n$ are coprime positive inte... | 10. In $\triangle A B C$, $A B=\sqrt{30}, A C=\sqrt{6}, B C$ $=\sqrt{15}$, there is a point $D$ such that $A D$ bisects $B C$ and $\angle A D B$ is a right angle, the ratio $\frac{S_{\triangle A D B}}{S_{\triangle A B C}}$ can be written as $\frac{m}{n}$. ($m, n$ are coprime positive integers). Then $m+n=$ | "[\"65\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1581, "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:15575"}] | {
"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
} |
[
"532"
] | [
"skywork"
] | 1b7e38cc6aeeb447575b92da7a05fc922f07154948507dcd66744e96e5488811 | [
{
"content": "Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$ , or as a $b^{}_{}$ wh... | Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$ , or as a $b^{}_{}$ when it should be an $a^... | "[\"532\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 1584, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15576"}, {"dataset": "skywork", "extra_in... | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0\\leqslantx<y\\leqslant\\pi"
] | [
"skywork"
] | 1bc6f92e4ec0c7c334fa2d03adbffc679118c35060daf60b7fc1d154cae128b1 | [
{
"content": "9. If $x, y \\in[0, \\pi]$, for any real number $t$, the expression $\\left(-2 \\cos t-\\frac{1}{2} \\cos x \\cos y\\right) \\cos x \\cos y-1$ $-\\cos x+\\cos y-\\cos 2 t$ is always negative, find the relationship that $x, y$ must satisfy.",
"role": "user"
}
] | 9. If $x, y \in[0, \pi]$, for any real number $t$, the expression $\left(-2 \cos t-\frac{1}{2} \cos x \cos y\right) \cos x \cos y-1$ $-\cos x+\cos y-\cos 2 t$ is always negative, find the relationship that $x, y$ must satisfy. | "[\"0\\\\leqslantx<y\\\\leqslant\\\\pi\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1582, "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:15577"}] | {
"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
} |
[
"240"
] | [
"skywork"
] | c6ab8f1f57d6ce58813eb345f97578e4cc94820f86c226ace92d4e9cb32974a2 | [
{
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\).",
"role": "user"
}
] | The numbers \( a, b, c, d \) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | "[\"240\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1586, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "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:15578"}, {"dataset": "skywork", "extra_inf... | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"0.48"
] | [
"skywork"
] | 6b6140794d1ea9be7b828696c955c6e419bd301e3a64295daa3224cdc38fa6af | [
{
"content": "The math scores of a high school's college entrance examination, denoted as $\\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\\xi < 110) = 0.98$, find the value of $P(90 < \\xi < 100)$.",
"role": "user"
}
] | The math scores of a high school's college entrance examination, denoted as $\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\xi < 110) = 0.98$, find the value of $P(90 < \xi < 100)$. | "[\"0.48\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1580, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15579"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{5}{21}"
] | [
"skywork"
] | 7ae372cd7672a0557385fc0542527f511ca4639ea81aa4cd345dd71ff531e0ea | [
{
"content": "During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the \"Qingming Sacrifice to the Martyrs\" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students can only choose one... | During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the "Qingming Sacrifice to the Martyrs" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students can only choose one way to participate. It ... | "[\"\\\\frac{5}{21}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1554, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15580"}] | {
"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
} |
[
"36"
] | [
"skywork"
] | 6e1813135a89ed1da5c1f30183678b2661ee7b895b3c251923a58082de59d863 | [
{
"content": "If $x=4$ and $y=3 x$ and $z=2 y$, then the value of $y+z$ is\n(A) 12\n(B) 20\n(C) 40\n(D) 24\n(E) 36",
"role": "user"
}
] | If $x=4$ and $y=3 x$ and $z=2 y$, then the value of $y+z$ is
(A) 12
(B) 20
(C) 40
(D) 24
(E) 36 | "[\"36\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1599, "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:15581"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"5"
] | [
"skywork"
] | f5df2eb7fc5d624ccdaabad63749c3429e87a0de062833f600e5af327f74e3e0 | [
{
"content": "At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?",
"role": "user"
}
] | At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | "[\"5\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1596, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15582"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-1 / 4 < p \\leq 0 \\text{ or } p \\geq 2"
] | [
"skywork"
] | 05e9b85bbcbb6425265b41612e7849654a0ce4a15de26f309fdb78ae1bf95cc1 | [
{
"content": "6. (CZS 4) Find the real values of $p$ for which the equation \n$$ \\sqrt{2 p+1-x^{2}}+\\sqrt{3 x+p+4}=\\sqrt{x^{2}+9 x+3 p+9} $$ \nin $x$ has exactly two real distinct roots $(\\sqrt{t}$ means the positive square root of $t$ ).",
"role": "user"
}
] | 6. (CZS 4) Find the real values of $p$ for which the equation
$$ \sqrt{2 p+1-x^{2}}+\sqrt{3 x+p+4}=\sqrt{x^{2}+9 x+3 p+9} $$
in $x$ has exactly two real distinct roots $(\sqrt{t}$ means the positive square root of $t$ ). | "[\"-1 / 4 < p \\\\leq 0 \\\\text{ or } p \\\\geq 2\"]" | train-math-numinamath1.5_olympiads_ref | [{"dataset": "skywork", "extra_info": {"index": 1587, "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:15583"}] | {
"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
} |
[
"15625"
] | [
"skywork"
] | 5ddb32d0a1f37a4dd278e6ed01c9e5951782025b361b6177541cd9b4cc634d8d | [
{
"content": "Calculate $\\left(\\sqrt{(\\sqrt{5})^4}\\right)^6$.",
"role": "user"
}
] | Calculate $\left(\sqrt{(\sqrt{5})^4}\right)^6$. | "[\"15625\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1601, "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:15584"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"skywork"
] | c574be8f660dbe95a48075ab51495a99b57c27bceb10208d6df0a469f79547ca | [
{
"content": "13.384 Five people perform a certain job. The first, second, and third, working together, can complete the entire job in 327.5 hours; the first, third, and fifth together - in 5 hours; the first, third, and fourth together - in 6 hours; and the second, fourth, and fifth together - in 4 hours. In w... | 13.384 Five people perform a certain job. The first, second, and third, working together, can complete the entire job in 327.5 hours; the first, third, and fifth together - in 5 hours; the first, third, and fourth together - in 6 hours; and the second, fourth, and fifth together - in 4 hours. In what time interval will... | "[\"3\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1590, "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:15585"}] | {
"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
} |
[
"56"
] | [
"skywork"
] | 7f04872987a81d54298fe3249d550e12c9334010fea16fb94aa260afeed35e7b | [
{
"content": "A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number.",
"role": "user"
}
] | A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number. | "[\"56\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1614, "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:15586"}] | {
"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
} |
[
"3302"
] | [
"skywork"
] | cd7bdeecda6387ea09556c0a399fdb7240d139be332f1a81d65bf3f8e422f47e | [
{
"content": "24. Find the sum of all even factors of 1152 .",
"role": "user"
}
] | 24. Find the sum of all even factors of 1152 . | "[\"3302\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1592, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15587"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"14"
] | [
"skywork"
] | 396298753ff5d05301f8ffd7208c9520133fd374532555ba055c2a1610a905cf | [
{
"content": "At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the $3$ identical bags, assuming he might leave some of the bags empty?",
"role": "user"
}
] | At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought in the $3$ identical bags, assuming he might leave some of the bags empty? | "[\"14\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1605, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15588"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1"
] | [
"skywork"
] | 7e022e4c1146a460a40137d13aa9cf2260986b5ce301c3d75a3377a8142821f8 | [
{
"content": "A circle is inscribed in a right angle. A chord connecting the points of tangency is equal to 2. Find the distance from the center of the circle to this chord.\n\n#",
"role": "user"
}
] | A circle is inscribed in a right angle. A chord connecting the points of tangency is equal to 2. Find the distance from the center of the circle to this chord.
# | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1595, "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:15589"}] | {
"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
} |
[
"6044"
] | [
"skywork"
] | 9f595bbeeaeae30f0c0b9b95c5238efaa4bd39b1ca3983a23513d7c957de0668 | [
{
"content": "Given the function $f(x) = \\frac{x+3}{x^2+1}$, and $g(x) = x - \\ln(x-p)$.\n(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\\left(\\frac{1}{3}, f\\left(\\frac{1}{3}\\right)\\right)$;\n(II) Determine the number of zeros of the function $g(x)$, and explain the reason... | Given the function $f(x) = \frac{x+3}{x^2+1}$, and $g(x) = x - \ln(x-p)$.
(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\left(\frac{1}{3}, f\left(\frac{1}{3}\right)\right)$;
(II) Determine the number of zeros of the function $g(x)$, and explain the reason;
(III) It is known that the seq... | "[\"6044\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1568, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 7, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15590"}] | {
"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
} |
[
"93"
] | [
"skywork"
] | 930c56892ed5f5f4695767999863a1f1fe8ce2e7da1266485f552ce9ce2ed054 | [
{
"content": "16. Let $m$ and $n$ be positive integers satisfying\n$$\nm n^{2}+876=4 m n+217 n \\text {. }\n$$\n\nFind the sum of all possible values of $m$.",
"role": "user"
}
] | 16. Let $m$ and $n$ be positive integers satisfying
$$
m n^{2}+876=4 m n+217 n \text {. }
$$
Find the sum of all possible values of $m$. | "[\"93\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1593, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 7, "DeepSeek-R1-Distill-Qwen-7B": 10}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15591"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{3}{5}"
] | [
"skywork"
] | 8c508dd4218a19a16ecc5edb9e13c1b5fe417652b1845aca04ee284129fc399a | [
{
"content": "4. In Rt $\\triangle A B C$, $C D$ is the altitude on the hypotenuse $A B$, the three sides $a, b, c$ of $\\triangle A B C$ are all positive integers, $B D$ $=27$. Then $\\cos B=$ $\\qquad$",
"role": "user"
}
] | 4. In Rt $\triangle A B C$, $C D$ is the altitude on the hypotenuse $A B$, the three sides $a, b, c$ of $\triangle A B C$ are all positive integers, $B D$ $=27$. Then $\cos B=$ $\qquad$ | "[\"\\\\frac{3}{5}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1578, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 6, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 1}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15592"}] | {
"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
} |
[
"1594"
] | [
"skywork"
] | 2fdf06b87d08ac7bc2d301a2856f8cfad939441bd82d22125ab03a07fa32f9cb | [
{
"content": "Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, and $a+b+c$ be 7 distinct prime numbers, and suppose that the sum of two of $a$, $b$, and $c$ is 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$.\n(Liang Darong, problem c... | Let $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$, and $a+b+c$ be 7 distinct prime numbers, and suppose that the sum of two of $a$, $b$, and $c$ is 800. Let $d$ be the difference between the largest and smallest of these 7 prime numbers. Find the maximum possible value of $d$.
(Liang Darong, problem contributor) | "[\"1594\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1602, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15593"}] | {
"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
} |
[
"f(kp + r) = \\epsilon(r) (kp + r)"
] | [
"skywork"
] | 9103ffddd5de777f62486004f24ac3a38b0335cef588a1523633eaf0e7b90bcd | [
{
"content": "Find all functions $f: \\mathbb{N} \\rightarrow \\mathbb{Z}$ such that $|f(k)| \\leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions:\n\n$1)$ For all $a \\in \\mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$\n$2)$ For all $a \\in \... | Find all functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that $|f(k)| \leq k$ for all positive integers $k$ and there is a prime number $p>2024$ which satisfies both of the following conditions:
$1)$ For all $a \in \mathbb{N}$ we have $af(a+p) = af(a)+pf(a),$
$2)$ For all $a \in \mathbb{N}$ we have $p|a^{\frac{p... | "[\" f(kp + r) = \\\\epsilon(r) (kp + r) \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1589, "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:15594"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"0.297"
] | [
"skywork"
] | 82ff54e8c5fe54aa2d16f70d3b87999531ed5651509471103c624c363ca19fbf | [
{
"content": "What is the difference between the largest and smallest numbers in the list $0.023,0.302,0.203,0.320,0.032$?",
"role": "user"
}
] | What is the difference between the largest and smallest numbers in the list $0.023,0.302,0.203,0.320,0.032$? | "[\"0.297\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1628, "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:15595"}] | {
"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
} |
[
"10000"
] | [
"skywork"
] | 381ca586ccdbd2f1408ea93f211fa60a9684fe6ed608dda085a6534036a34eca | [
{
"content": "The sum of the first 100 positive integers is 5050 . That is, $1+2+\\cdots+99+100=5050$. What is the sum of the first 100 positive odd integers?\n(A) 5050\n(B) 10000\n(C) 10050\n(D) 10100\n(E) 10150",
"role": "user"
}
] | The sum of the first 100 positive integers is 5050 . That is, $1+2+\cdots+99+100=5050$. What is the sum of the first 100 positive odd integers?
(A) 5050
(B) 10000
(C) 10050
(D) 10100
(E) 10150 | "[\"10000\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1600, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15596"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"64"
] | [
"skywork"
] | 1a399ee7b84c5c93eafbb5ba3ef32e43beec63ef787df171e652bfd4451c7506 | [
{
"content": "Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\\left( 1+\\frac1x\\right)\\times\\left(1+\\frac1y\\right)\\times\\left(1 +\\frac1z\\right)\\ge 64$\nWhen does equality hold?",
"role": "user"
}
] | Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$
When does equality hold? | "[\" 64 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 1613, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "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:15597"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{1}{4}"
] | [
"skywork"
] | 43e99880146d99cb7ee7c128e4911c5eb783e3fdc1238b991104960523ed5c4e | [
{
"content": "5. Petya writes on a piece of paper a string of 5 zeros and 19 ones, arranged in a completely random order. Find the mathematical expectation of the random variable - the number of zeros written before the first one appears.",
"role": "user"
}
] | 5. Petya writes on a piece of paper a string of 5 zeros and 19 ones, arranged in a completely random order. Find the mathematical expectation of the random variable - the number of zeros written before the first one appears. | "[\"\\\\frac{1}{4}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1611, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15598"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"3-\\sqrt{3}"
] | [
"skywork"
] | 5181e304c903d5782012fc4964e207ac0504e1e8806a0d8b36693926f38a0d6f | [
{
"content": "24. A square has its vertices on the edges of a regular hexagon. Two of the edges of the square are parallel to two edges of the hexagon, as shown in the diagram. The sides of the hexagon have length 1 .\nWhat is the length of the sides of the square?",
"role": "user"
}
] | 24. A square has its vertices on the edges of a regular hexagon. Two of the edges of the square are parallel to two edges of the hexagon, as shown in the diagram. The sides of the hexagon have length 1 .
What is the length of the sides of the square? | "[\"3-\\\\sqrt{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1618, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 11, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15599"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"-8"
] | [
"skywork"
] | 06d2110fe221fed3b195519beb2c7c0be577dfb83759c69aa09aa73bb5012f27 | [
{
"content": "If $3(-2)=\\nabla+2$, then $\\nabla$ equals\n(A) -2\n(B) 0\n(C) -8\n(D) -6\n(E) -4",
"role": "user"
}
] | If $3(-2)=\nabla+2$, then $\nabla$ equals
(A) -2
(B) 0
(C) -8
(D) -6
(E) -4 | "[\"-8\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1620, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 0, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15600"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1/7"
] | [
"skywork"
] | e3deda670ac36e084bf23915c329dc36fd81086d2c46ed41a3eb8e7a090f8b18 | [
{
"content": "In the complex plane, the vertices of the right-angled triangle $\\triangle ABC$ correspond to the complex numbers $z+1$, $2z+1$, and $(z+1)^2$, with $A$ being the right-angled vertex, and $|z|=2$. Define the set $M=\\left\\{m \\mid z^{m} \\in \\mathbf{R}, m \\in \\mathbf{N}_{+}\\right\\}$ and set... | In the complex plane, the vertices of the right-angled triangle $\triangle ABC$ correspond to the complex numbers $z+1$, $2z+1$, and $(z+1)^2$, with $A$ being the right-angled vertex, and $|z|=2$. Define the set $M=\left\{m \mid z^{m} \in \mathbf{R}, m \in \mathbf{N}_{+}\right\}$ and set $P=\left\{x \mid x=\frac{1}{2^{... | "[\"1/7\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 1607, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15601"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{7}{12}"
] | [
"skywork"
] | ddf72f187387aec1af95ed0a4d81fb0a603e17eee9e59b9af14ceaf4f6516ee7 | [
{
"content": "5. In $\\triangle A B C$, $A B=3, A C=4, B C=5, I$ is the incenter of $\\triangle A B C$, and $P$ is a point within $\\triangle I B C$ (including the boundary). If $\\overrightarrow{A P}=\\lambda \\overrightarrow{A B}+\\mu \\overrightarrow{A C}(\\lambda ; \\mu \\in \\mathbf{R})$, then the minimum ... | 5. In $\triangle A B C$, $A B=3, A C=4, B C=5, I$ is the incenter of $\triangle A B C$, and $P$ is a point within $\triangle I B C$ (including the boundary). If $\overrightarrow{A P}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}(\lambda ; \mu \in \mathbf{R})$, then the minimum value of $\lambda+\mu$ is | "[\"\\\\frac{7}{12}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 1603, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:15602"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2"
] | [
"skywork"
] | 438be92451257db2e86902dc00f5f2894cabcf93b64360e07fdd8f6264180b35 | [
{
"content": "4. Calculate: $\\frac{2008+2007 \\times 2009}{2008 \\times 2009-1}+\\frac{2009+2008 \\times 2010}{2009 \\times 2010-1}=$",
"role": "user"
}
] | 4. Calculate: $\frac{2008+2007 \times 2009}{2008 \times 2009-1}+\frac{2009+2008 \times 2010}{2009 \times 2010-1}=$ | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 1619, "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:15603"}, {"dataset": "skywork", "extra_inf... | {
"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
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.