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 |
|---|---|---|---|---|---|---|---|---|
[
"\\textbf{(D)}"
] | [
"skywork"
] | 391017faa711fffe91a156d157cf273b0ca5b3afe497eda73249a07d00561ae9 | [
{
"content": "The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\\times2!+a_3\\times3!+ \\ldots a_n \\times n!$ \nwhere $a_1, a_2, a_3 ... a_n$ are integers such that $0 \\le a_k \\le k,$ and $n!$ means $n(n-1)(n-2)...2 \\times 1$. Find $a_4$\n$\\textbf{(A)}\\ 0\\qquad ... | The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$
where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k,$ and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}... | "[\"\\\\textbf{(D)}\"]" | train-math-numinamath1.5_amc_aime | [{"dataset": "skywork", "extra_info": {"index": 2569, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "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:16531"}] | {
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[
"942"
] | [
"skywork"
] | 2e74a9c8a28698c2cf36cdc8531a6d07c8a54f750d0f04c9c1f76e81d8a3a298 | [
{
"content": "Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\\lfloor K \\rfloor$.",
"ro... | Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$. | "[\"942\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2576, "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:16532"}] | {
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[
"46"
] | [
"skywork"
] | cc6450c1ea8ca5dd77badae39e60d27aef447dbd723474232c3903c0de65235e | [
{
"content": "In convex hexagon $ABCDEF$, all six sides are congruent, $\\angle A$ and $\\angle D$ are right angles, and $\\angle B, \\angle C, \\angle E,$ and $\\angle F$ are congruent. The area of the hexagonal region is $2116(\\sqrt{2}+1).$ Find $AB$.",
"role": "user"
}
] | In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B, \angle C, \angle E,$ and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1).$ Find $AB$. | "[\"46\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2570, "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:16533"}, {"dataset": "skywork", "extra_in... | {
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"reference_present": true
} |
[
"22"
] | [
"skywork"
] | 435dc759fa8cc8bf4c550589330359daaa18bb4b4817be0520afb6d7c3fc9e4e | [
{
"content": "3. Fill the numbers $1,2,3, \\ldots, 9,10$ into 10 circles that form a rectangle, such that the sum of the numbers on each side of the rectangle is equal. The maximum sum is ( ).",
"role": "user"
}
] | 3. Fill the numbers $1,2,3, \ldots, 9,10$ into 10 circles that form a rectangle, such that the sum of the numbers on each side of the rectangle is equal. The maximum sum is ( ). | "[\"22\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2575, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16534"}] | {
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} |
[
"35"
] | [
"skywork"
] | a5ffd5bd31fff2e270d1152cca492225dca3bbf896e1edb0ad0091ddafece716 | [
{
"content": "Example 4.9 Find the number of integer points (points with both coordinates as integers) within the triangle (including the boundary) formed by the lines $x+3y=12$, $x=0$, and $y=0$.\n\n untranslated text:\n求由直线 $x+3 y=12$, 直线 $x=0$ 及直线 $y=0$ 所围成的三角形内(包括边界)的整点(横坐标和纵坐标均是整数的点)的个数. \n\ntranslated tex... | Example 4.9 Find the number of integer points (points with both coordinates as integers) within the triangle (including the boundary) formed by the lines $x+3y=12$, $x=0$, and $y=0$.
untranslated text:
求由直线 $x+3 y=12$, 直线 $x=0$ 及直线 $y=0$ 所围成的三角形内(包括边界)的整点(横坐标和纵坐标均是整数的点)的个数.
translated text:
Find the number of integ... | "[\"35\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2577, "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:16535"}] | {
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} |
[
"n = 19"
] | [
"skywork"
] | d5ad63e7222b1a7c700b56d67a02716fa12d9359f9a59d5240a2328a08113c85 | [
{
"content": "A finite sequence of integers $ a_0, a_1, \\ldots, a_n$ is called quadratic if for each $ i$ in the set $ \\{1,2 \\ldots, n\\}$ we have the equality $ |a_i \\minus{} a_{i\\minus{}1}| \\equal{} i^2.$\r\n\r\na.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quad... | A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$
a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{... | "[\" n = 19 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2572, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16536"}] | {
"all_reference_components_parse": true,
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"reference_present": true
} |
[
"11"
] | [
"skywork"
] | e7f9d837eea1d7c0704185f29010c77ae161bbaedcedffc9da79ab7e4454ca56 | [
{
"content": "55 In a geometric sequence $\\left\\{a_{n}\\right\\}$, the first term $a_{1}=2^{-5}$, and the geometric mean of its first 11 terms is $2^{5}$. If, among the first 11 terms, one term is removed and the geometric mean of the remaining terms is $2^{4}$, then the removed term is\nA. The 8th term\nB. T... | 55 In a geometric sequence $\left\{a_{n}\right\}$, the first term $a_{1}=2^{-5}$, and the geometric mean of its first 11 terms is $2^{5}$. If, among the first 11 terms, one term is removed and the geometric mean of the remaining terms is $2^{4}$, then the removed term is
A. The 8th term
B. The 9th term
C. The 10th term... | "[\"11\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2571, "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:16537"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
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"reference_components": 1,
"reference_present": true
} |
[
"7"
] | [
"skywork"
] | 25920ab4232d2ff0cb298e805435556bd281b27e2d6b85a87a72708f6e12840b | [
{
"content": "## Task B-1.2.\n\nLet $n$ be a natural number and let\n\n$$\nA=0.1^{n} \\cdot 0.01^{n} \\cdot 0.001^{n} \\cdot 0.0001^{n} \\cdot \\ldots \\cdot 0 . \\underbrace{00 \\ldots 0}_{199 \\text { zeros }} 1^{n} .\n$$\n\nThe number $A^{-1}$ has 140701 digits. Determine the number $n$.",
"role": "user"... | ## Task B-1.2.
Let $n$ be a natural number and let
$$
A=0.1^{n} \cdot 0.01^{n} \cdot 0.001^{n} \cdot 0.0001^{n} \cdot \ldots \cdot 0 . \underbrace{00 \ldots 0}_{199 \text { zeros }} 1^{n} .
$$
The number $A^{-1}$ has 140701 digits. Determine the number $n$. | "[\"7\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2568, "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:16538"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{3}"
] | [
"skywork"
] | 858bf60bdd26932581c170fb6b962891599e3ac21ccd7dfe3cab328cbc702ac8 | [
{
"content": "A tetrahedron \\( P-ABC \\) has edge lengths \\( PA = BC = \\sqrt{6} \\), \\( PB = AC = \\sqrt{8} \\), and \\( PC = AB = \\sqrt{10} \\). Find the radius of the circumsphere of this tetrahedron.",
"role": "user"
}
] | A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron. | "[\"\\\\sqrt{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2588, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16539"}] | {
"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_{1}=\\pi+2k\\pi,x_{2}=\\frac{\\pi}{4}+k\\pi"
] | [
"skywork"
] | 1ae92ba12f738534c9ef999a622b9d985560697cdb02e8d966ab6a71f46029d5 | [
{
"content": "## Task B-4.1.\n\nSolve the equation\n\n$$\n\\sin x - \\cos x + \\tan x = 1\n$$",
"role": "user"
}
] | ## Task B-4.1.
Solve the equation
$$
\sin x - \cos x + \tan x = 1
$$ | "[\"x_{1}=\\\\pi+2k\\\\pi,x_{2}=\\\\frac{\\\\pi}{4}+k\\\\pi\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2600, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16541"}] | {
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"reference_components": 1,
"reference_present": true
} |
[
"30"
] | [
"skywork"
] | 567b646d5ca49fd207c81da7f2d3bb4efc37b93d61e8f65b1757cf25b998d269 | [
{
"content": "18. Gargamel is designing a trap to catch the Smurfs. He plans to use 56 small squares, each with an area of 1 square decimeter, to form a large rectangular trap. The minimum perimeter of the trap he can make is $\\qquad$ decimeters.",
"role": "user"
}
] | 18. Gargamel is designing a trap to catch the Smurfs. He plans to use 56 small squares, each with an area of 1 square decimeter, to form a large rectangular trap. The minimum perimeter of the trap he can make is $\qquad$ decimeters. | "[\"30\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2579, "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:16542"}] | {
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} |
[
"66"
] | [
"skywork"
] | dd5c5d21a2033e7ca0e45db07d55f0c9918e328da9cbb94927a38b97714a6994 | [
{
"content": "1. On January 1, 2013, a little boy was given a bag of chocolate candies, containing 300 candies. Each day, the little boy ate one candy. On Sundays, Karlson would fly over, and the little boy would treat him to a couple of candies. How many candies did Karlson eat? (January 1, 2013, was a Tuesday... | 1. On January 1, 2013, a little boy was given a bag of chocolate candies, containing 300 candies. Each day, the little boy ate one candy. On Sundays, Karlson would fly over, and the little boy would treat him to a couple of candies. How many candies did Karlson eat? (January 1, 2013, was a Tuesday).
2. Petya can swap ... | "[\"66\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2551, "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:16543"}] | {
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} |
[
"44"
] | [
"skywork"
] | 472ed320e525143ffa45310ed65dfe0d50e8b347bdd24cf5d161b239604d7223 | [
{
"content": "11.3. How many points (different from the origin) are there on the parabola $y=x^{2}$, such that the tangent at these points intersects both coordinate axes at points with integer coordinates not exceeding 2020 in absolute value?",
"role": "user"
}
] | 11.3. How many points (different from the origin) are there on the parabola $y=x^{2}$, such that the tangent at these points intersects both coordinate axes at points with integer coordinates not exceeding 2020 in absolute value? | "[\"44\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2594, "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:16544"}] | {
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"plain_numeric_reference": true,
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"reference_present": true
} |
[
"\\frac{36\\pi}{5}"
] | [
"skywork"
] | bd97a0d48e9d4d08e62e566ff29bb8857972ec798b4097a91bfca4c312da97d6 | [
{
"content": "3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\\overrightarrow{D P} \\cdot \\overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\\qquad$.",
... | 3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\overrightarrow{D P} \cdot \overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\qquad$. | "[\"\\\\frac{36\\\\pi}{5}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2573, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 10, "DeepSeek-R1-Distill-Qwen-7B": 7}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16545"}] | {
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"reference_present": true
} |
[
"2\\sqrt{6}"
] | [
"skywork"
] | cba0f400107af13bfeb07bad1136e70be7d75ba80d4742f975de77ca470b9e0f | [
{
"content": "A square sheet of paper has an area of $12 \\text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in si... | A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form. | "[\"2\\\\sqrt{6}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2591, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 9}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16546"}] | {
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} |
[
"\\sqrt{25+12 \\sqrt{3}}"
] | [
"skywork"
] | 4dba70f136473a37e9b16d286e134ea8d55e74a4ac3399ef28dde0a759977fd8 | [
{
"content": "II. (40 points) Let positive real numbers $a, b, c$ satisfy\n$$\n\\begin{array}{l}\na+b=\\sqrt{a b+9}, \\\\\nb+c=\\sqrt{b c+16}, \\\\\nc+a=\\sqrt{c a+25} .\n\\end{array}\n$$\n\nFind $a+b+c$.",
"role": "user"
}
] | II. (40 points) Let positive real numbers $a, b, c$ satisfy
$$
\begin{array}{l}
a+b=\sqrt{a b+9}, \\
b+c=\sqrt{b c+16}, \\
c+a=\sqrt{c a+25} .
\end{array}
$$
Find $a+b+c$. | "[\"\\\\sqrt{25+12 \\\\sqrt{3}}\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2593, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "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:16548"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2 \\sqrt {2}"
] | [
"skywork"
] | d9098f6c1ad0d0313835500351e6ce313a06d96d54ff12bad5b11767ccfb54d3 | [
{
"content": "Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\\neq 1$, which is an odd function defined on $\\mathbb{R}$.\n1. Find the value of $k$.\n2. If $f(1)= \\frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\\infty)$ is $-6$, find the value of $m$.",
"role"... | Given the function $f(x)=a^{x}-(k+1)a^{-x}$ where $a > 0$ and $a\neq 1$, which is an odd function defined on $\mathbb{R}$.
1. Find the value of $k$.
2. If $f(1)= \frac {3}{2}$, and the minimum value of $g(x)=a^{2x}+a^{-2x}-2mf(x)$ on $[0,+\infty)$ is $-6$, find the value of $m$. | "[\"2 \\\\sqrt {2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2585, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "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:16549"}] | {
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[
"712"
] | [
"skywork"
] | e22989fd82cf932dc739dd4e8fdcccf860d38e4bb903077f129a061879e2fd66 | [
{
"content": "## 4. Imagined Number\n\nGrga thought of a three-digit number, and his friends are trying to guess it.\n\nThese are their attempts:\n\nBoris: 218 Robert: $571 \\quad$ Marko: $732 \\quad$ Darko: 853\n\nGrga told them: \"One of you has guessed all the digits, and the others have only one each, but n... | ## 4. Imagined Number
Grga thought of a three-digit number, and his friends are trying to guess it.
These are their attempts:
Boris: 218 Robert: $571 \quad$ Marko: $732 \quad$ Darko: 853
Grga told them: "One of you has guessed all the digits, and the others have only one each, but none of the guessed digits are in ... | "[\"712\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2587, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "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:16550"}] | {
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"reference_components": 1,
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[
"329"
] | [
"skywork"
] | ffa1b413fa2a603a4a9dde7af9b80a98170f55f094a1d55af43cdc350ae69c3d | [
{
"content": "For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\\cdots+c_{210}$?",
"role": "user"
}
] | For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$? | "[\"329\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2596, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 4, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16551"}] | {
"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
} |
[
"2015"
] | [
"skywork"
] | 0f6203c80a781006e66f965885f0888795a01d9dade0834fff1ab3e3fdbd1fcc | [
{
"content": "12. For any set $S$, use $|S|$ to denote the number of elements in set $S$, and use $n(S)$ to denote the number of subsets of set $S$. If $A, B, C$ are three finite sets, and satisfy\n(1) $|A|=|B|=2016$;\n(2) $n(A)+n(B)+n(C)=n(A \\cup B \\cup C)$.\nThen the maximum value of $|A \\cap B \\cap C|$ i... | 12. For any set $S$, use $|S|$ to denote the number of elements in set $S$, and use $n(S)$ to denote the number of subsets of set $S$. If $A, B, C$ are three finite sets, and satisfy
(1) $|A|=|B|=2016$;
(2) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$.
Then the maximum value of $|A \cap B \cap C|$ is $\qquad$ | "[\"2015\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2590, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 7, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16552"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"814\\mathrm{~}"
] | [
"skywork"
] | d2c51b8f6db7a6984804d0bc3ce5e3220a003f0fcf5288d5076508c05add3c4a | [
{
"content": "## Task 3 - V10713\n\nFor an electrical line of $7 \\mathrm{~km}$ in length, $259 \\mathrm{~kg}$ of copper wire is required.\n\nHow many kilograms of copper wire of the same thickness are required for a line of $22 \\mathrm{~km}$ in length?",
"role": "user"
}
] | ## Task 3 - V10713
For an electrical line of $7 \mathrm{~km}$ in length, $259 \mathrm{~kg}$ of copper wire is required.
How many kilograms of copper wire of the same thickness are required for a line of $22 \mathrm{~km}$ in length? | "[\"814\\\\mathrm{~}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2609, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "DeepSeek-R1-Distill-Qwen-32B": 5, "DeepSeek-R1-Distill-Qwen-7B": 5}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16553"}] | {
"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
} |
[
"365"
] | [
"skywork"
] | e0c51f3cb5e025ea985d7843c70cdc8143ce4e9d2d7200b5579b174e5b71c5aa | [
{
"content": "Seven students count from 1 to 1000 as follows:\nAlice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.\nBarbara says all of the numbers that Alice doesn't say, except she also skips th... | Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each ... | "[\"365\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2597, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 11}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16554"}] | {
"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
} |
[
"7"
] | [
"skywork"
] | 9e7057a053672633b20f37e6918471cc68e5394d8ecb970721a4646b32daac59 | [
{
"content": "Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\\angle DAC = 30^\\circ$ and $m\\angle BAC = 45^\\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in terms of $\\pi$ as $\\... | Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 30^\circ$ and $m\angle BAC = 45^\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, whe... | "[\"7\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2602, "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:16555"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"3"
] | [
"skywork"
] | 0ceb1e0231cd8d03dd332b5dbfb444a3d8cf1261996a424d661c166f70922db1 | [
{
"content": "Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?\n$\\textbf{(A)}\\ 3 \\qquad \\textbf{(B)}\\ 4 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 9$",
"role": "user"
}
] | Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$ | "[\"3\"]" | train-math-numinamath1.5_amc_aime | [{"dataset": "skywork", "extra_info": {"index": 2601, "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:16556"}] | {
"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{257}{16}"
] | [
"skywork"
] | 9d5a15b639683d90b2bc3e6e80b32e8cac6ebaf86149a9340f8a1386162ac63d | [
{
"content": "5. $[\\mathbf{5}]$ Find all real values of $x$ for which\n$$\n\\frac{1}{\\sqrt{x}+\\sqrt{x-2}}+\\frac{1}{\\sqrt{x+2}+\\sqrt{x}}=\\frac{1}{4} .\n$$",
"role": "user"
}
] | 5. $[\mathbf{5}]$ Find all real values of $x$ for which
$$
\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}}=\frac{1}{4} .
$$ | "[\"\\\\frac{257}{16}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2581, "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:16557"}] | {
"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
} |
[
"333"
] | [
"skywork"
] | 7425b66b6287637f0a77ae049ef373bb0308349e2f5356c40ad28c6d68d322be | [
{
"content": "6. The reading on a factory's electricity meter is 52222 kilowatts. After several days, the meter reading (a five-digit number) again shows four identical digits. How much electricity, in kilowatts, did the factory use at least in these days?\n\n untranslated portion: $\\qquad$",
"role": "user... | 6. The reading on a factory's electricity meter is 52222 kilowatts. After several days, the meter reading (a five-digit number) again shows four identical digits. How much electricity, in kilowatts, did the factory use at least in these days?
untranslated portion: $\qquad$ | "[\"333\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2604, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16558"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"(4;1)"
] | [
"skywork"
] | 171b1389af8b52aa8e873b2a82862fa667e278aece7d2ff45adb61174486c455 | [
{
"content": "3. Solve the system of equations $\\left\\{\\begin{array}{l}x^{2} y-x y^{2}-5 x+5 y+3=0, \\\\ x^{3} y-x y^{3}-5 x^{2}+5 y^{2}+15=0 .\\end{array}\\right.$",
"role": "user"
}
] | 3. Solve the system of equations $\left\{\begin{array}{l}x^{2} y-x y^{2}-5 x+5 y+3=0, \\ x^{3} y-x y^{3}-5 x^{2}+5 y^{2}+15=0 .\end{array}\right.$ | "[\"(4;1)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2616, "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:16559"}, {"dataset": "skywork", "extra_inf... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"32"
] | [
"skywork"
] | 745a5efec5f8ee1f8657b0214ec723ec7f0de01728b4d4ba87e2edbf70bb0a7e | [
{
"content": "Given that the general term of the sequence $\\{a_{n}\\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\\{{a}_{n}{a}_{n+1}{a}_{n+2}\\}(n∈{N}^{*})$ reaches its maximum value.",
"role": "user"
}
] | Given that the general term of the sequence $\{a_{n}\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\{{a}_{n}{a}_{n+1}{a}_{n+2}\}(n∈{N}^{*})$ reaches its maximum value. | "[\"32\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2595, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 3, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16560"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"12.5"
] | [
"skywork"
] | fc93db76c5bbc1ca97ef8e1dc52b0680cb2a4a1876978dd37798cff7d723fab6 | [
{
"content": "6. A young sprinter ran 90 meters with the wind in 10 seconds. Under the same wind speed, he ran 70 meters against the wind in 10 seconds. Question: In the absence of wind, how many seconds would it take him to run 100 meters?",
"role": "user"
}
] | 6. A young sprinter ran 90 meters with the wind in 10 seconds. Under the same wind speed, he ran 70 meters against the wind in 10 seconds. Question: In the absence of wind, how many seconds would it take him to run 100 meters? | "[\"12.5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2620, "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:16561"}] | {
"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
} |
[
"15"
] | [
"skywork"
] | ed9b35715ef4bc1640964b693830caf28d48c5bbbbdf2b13b6146c15107ba23f | [
{
"content": "Each cell of a $29 \\times 29$ table contains one of the integers $1, 2, 3, \\ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central c... | Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table. | "[\"15\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2624, "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:16562"}] | {
"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
} |
[
"135\\mathrm{~}"
] | [
"skywork"
] | b0c2e9381a86ba0ce6b4bef78cf5a0eaddd07afea7f5bb014750047a74029838 | [
{
"content": "## 62. Math Puzzle $7 / 70$\n\nTwo trains pass each other in opposite directions. One travels at a speed of $36 \\frac{\\mathrm{km}}{\\mathrm{h}}$, the other at $45 \\frac{\\mathrm{km}}{\\mathrm{h}}$. A passenger in the second train noticed that the first train took 6 s to pass him.\n\nHow long wa... | ## 62. Math Puzzle $7 / 70$
Two trains pass each other in opposite directions. One travels at a speed of $36 \frac{\mathrm{km}}{\mathrm{h}}$, the other at $45 \frac{\mathrm{km}}{\mathrm{h}}$. A passenger in the second train noticed that the first train took 6 s to pass him.
How long was the train? | "[\"135\\\\mathrm{~}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2614, "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:16563"}] | {
"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"
] | b2816d2e64b442e5b22abaf8b50a201c134d728432708387ff50cf115b50a8e7 | [
{
"content": "6. If $x, y$ are real numbers, then the minimum value of the maximum number among $|2 x+y|,|x-y|,|1+y|$ is\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.",
"role": "user"
}
] | 6. If $x, y$ are real numbers, then the minimum value of the maximum number among $|2 x+y|,|x-y|,|1+y|$ is
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | "[\"\\\\frac{1}{2}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2619, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 3}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16564"}] | {
"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"
] | 1a941d872dffc15250b138993707ec1ea6fde574575cbe2857bb07bd7cc873cb | [
{
"content": "Find the smallest positive integer $n$, such that $x_{1}{ }^{3}+x_{2}{ }^{3}+\\cdots+x_{n}{ }^{3}=2002^{2002}$ has integer solutions.",
"role": "user"
}
] | Find the smallest positive integer $n$, such that $x_{1}{ }^{3}+x_{2}{ }^{3}+\cdots+x_{n}{ }^{3}=2002^{2002}$ has integer solutions. | "[\"4\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2607, "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:16565"}] | {
"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
} |
[
"0.5\\mathrm{~L}"
] | [
"skywork"
] | cf70cce893bf0e6a5290b78e8da5dffb3221d05861cac9a5a977327f07657bb2 | [
{
"content": "Water is poured from a full $1.5 \\mathrm{~L}$ bottle into an empty glass until both the glass and the bottle are $\\frac{3}{4}$ full. What is the volume of the glass?\n(A) $0.5 \\mathrm{~L}$\n(B) $0.75 \\mathrm{~L}$\n(C) $1.125 \\mathrm{~L}$\n(D) $0.6 \\mathrm{~L}$\n(E) $0.4 \\mathrm{~L}$",
"... | Water is poured from a full $1.5 \mathrm{~L}$ bottle into an empty glass until both the glass and the bottle are $\frac{3}{4}$ full. What is the volume of the glass?
(A) $0.5 \mathrm{~L}$
(B) $0.75 \mathrm{~L}$
(C) $1.125 \mathrm{~L}$
(D) $0.6 \mathrm{~L}$
(E) $0.4 \mathrm{~L}$ | "[\"0.5\\\\mathrm{~L}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2626, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 13, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16566"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2 \\times 1008!"
] | [
"skywork"
] | 6f8cb6533dddc98cdbf566f002bee4a29ab8ff9c2e68c37e3aa0c217764da7e2 | [
{
"content": "A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the poly... | A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of... | "[\"2 \\\\times 1008!\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2606, "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:16569"}, {"dataset": "skywork", "extra_... | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"13"
] | [
"skywork"
] | ed6da2e1f02ae6221e095fc01ff718c268d38a9d388c9a3cd6186535fc5237cc | [
{
"content": "Positive real numbers \\( x, y, z \\) satisfy\n\\[\n\\left\\{\n\\begin{array}{l}\n\\frac{2}{5} \\leqslant z \\leqslant \\min \\{x, y\\}, \\\\\nxz \\geqslant \\frac{4}{15}, \\\\\nyz \\geqslant \\frac{1}{5}.\n\\end{array}\n\\right.\n\\]\n\nFind the maximum value of \\( \\frac{1}{x} + \\frac{2}{y} + ... | Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]
Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \). | "[\"13\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2603, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 2}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16570"}] | {
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"reference_components": 1,
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} |
[
"1"
] | [
"skywork"
] | 96f8a925ed93292a61b690f0628730837ba308fd56849d323eda29dc63728b67 | [
{
"content": "Let $F(0) = 0,$ $F(1) = \\frac{3}{2},$ and\n\\[F(n) = \\frac{5}{2} F(n - 1) - F(n - 2)\\]for $n \\ge 2.$ Find\n\\[\\sum_{n = 0}^\\infty \\frac{1}{F(2^n)}.\\]",
"role": "user"
}
] | Let $F(0) = 0,$ $F(1) = \frac{3}{2},$ and
\[F(n) = \frac{5}{2} F(n - 1) - F(n - 2)\]for $n \ge 2.$ Find
\[\sum_{n = 0}^\infty \frac{1}{F(2^n)}.\] | "[\"1\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2610, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:16571"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"-\\frac {1}{5}"
] | [
"skywork"
] | bbd60a3d860328b00c10560221ccc7e0e31a62514dd4cd2ce4765ce49946d03c | [
{
"content": "Given two lines $l_1: y = 2x$, $l_2: y = -2x$, and a line $l$ passing through point $M(-2, 0)$ intersects $l_1$ and $l_2$ at points $A$ and $B$, respectively, where point $A$ is in the third quadrant, point $B$ is in the second quadrant, and point $N(1, 0)$; \n(1) If the area of $\\triangle NAB$ ... | Given two lines $l_1: y = 2x$, $l_2: y = -2x$, and a line $l$ passing through point $M(-2, 0)$ intersects $l_1$ and $l_2$ at points $A$ and $B$, respectively, where point $A$ is in the third quadrant, point $B$ is in the second quadrant, and point $N(1, 0)$;
(1) If the area of $\triangle NAB$ is 16, find the equation... | "[\"-\\\\frac {1}{5}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2598, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "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:16572"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"112"
] | [
"skywork"
] | 817d9d9e184ec5b18ba64576bd87adb00b3535187947ad101010e0d431599c0e | [
{
"content": "The numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\\text{th}}$ term of the sequence is $\\log{b^n}$. What is $n$?",
"role": "user"
}
] | The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is $n$? | "[\"112\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2621, "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:16573"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"f(x) = \\frac{x^4 + x^2}{2}"
] | [
"skywork"
] | 02d19e57e17a7ceec9517257ee958f311eb8df41d3a3b21bf2b1a3f1750c2bd7 | [
{
"content": "Let $ f(x)$ be a polynomial with the following properties:\r\n(i) $ f(0)\\equal{}0$; (ii) $ \\frac{f(a)\\minus{}f(b)}{a\\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?",
... | Let $ f(x)$ be a polynomial with the following properties:
(i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers? | "[\" f(x) = \\\\frac{x^4 + x^2}{2} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2630, "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:16574"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\sqrt{3}"
] | [
"skywork"
] | c34a81c40e083a237ccfe851440edd54aa977475a6d4baaefcf1cb3367cc88d5 | [
{
"content": "A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola.",
"role": "user"
}... | A line is drawn through the left focus $F_1$ of a hyperbola at an angle of $30^{\circ}$, intersecting the right branch of the hyperbola at point P. If a circle with diameter PF_1 passes through the right focus of the hyperbola, calculate the eccentricity of the hyperbola. | "[\"\\\\sqrt{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2618, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "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:16575"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"840"
] | [
"skywork"
] | d6d5481c3f5f14bda7a181ba4a0777d858bb536d753eb000a3f24c90a6eafb78 | [
{
"content": "1.68 A convex polyhedron has a surface composed of 12 squares, 8 regular hexagons, and 6 regular octagons, and each vertex is the common vertex of one square, one octagon, and one hexagon. Among the line segments connecting the vertices of the polyhedron, how many are inside the polyhedron, and no... | 1.68 A convex polyhedron has a surface composed of 12 squares, 8 regular hexagons, and 6 regular octagons, and each vertex is the common vertex of one square, one octagon, and one hexagon. Among the line segments connecting the vertices of the polyhedron, how many are inside the polyhedron, and not on the faces or edge... | "[\"840\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2623, "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:16576"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{29}{120}"
] | [
"skywork"
] | 061335c10dfe7cbd1396b0c17666209b3bae1b8f4505afd2970f8d9139c1718a | [
{
"content": "At Pine Ridge Elementary School, one third of the students ride the school bus home. One fifth of the students are picked up by car. One eighth of the students go home on their skateboards. Another one tenth of the students share rides with classmates. The rest of the students walk home. What frac... | At Pine Ridge Elementary School, one third of the students ride the school bus home. One fifth of the students are picked up by car. One eighth of the students go home on their skateboards. Another one tenth of the students share rides with classmates. The rest of the students walk home. What fractional part of the stu... | "[\"\\\\frac{29}{120}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2615, "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:16577"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{6}{7}"
] | [
"skywork"
] | 4082427d50bdd2fb1bb31588a8c6cd90689b3a77f5a4138b353b3103bf48308a | [
{
"content": "Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk. How much milk did every cup contain,... | Seven elves are sitting at a round table. Each elf has a cup. Some cups are filled with some milk. Each elf in turn and clockwise divides all his milk between six other cups. After the seventh has done this, every cup was containing the initial amount of milk. How much milk did every cup contain, if there was three li... | "[\" \\\\frac{6}{7} \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2613, "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:16578"}] | {
"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
} |
[
"9100"
] | [
"skywork"
] | 67886bfe9001ae6ed8c2b874224c14d3b75716609ad0136ba487f0c85961c44c | [
{
"content": "For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible... | For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tourname... | "[\"9100\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2628, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:16579"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{13}{2550}"
] | [
"skywork"
] | 1218ffd4adde67885177941d264116685e5cddd371f9df597437e70a1a6cabcf | [
{
"content": "Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\clubsuit$, the second card is a $\\heartsuit$, and the third card is a king?",
"role": "user"
}
] | Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$, the second card is a $\heartsuit$, and the third card is a king? | "[\"\\\\frac{13}{2550}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2622, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16580"}] | {
"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
} |
[
"33"
] | [
"skywork"
] | 8a3e6fbd9eb2328cea56aa7b6c050067f3c11f4fa0276ceac474495658e3f7bb | [
{
"content": "A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears?",
"role": "user"
}
] | A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears? | "[\"33\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2632, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:16581"}, {"dataset": "skywork", "extra_inf... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"60"
] | [
"skywork"
] | 82b3df502898c18ec1c0379f2dfa221fccafa7a52003bbdc328b7c1e77acb816 | [
{
"content": "Consider the numbers $\\{24,27,55,64,x\\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ .",
"role": "user"
}
] | Consider the numbers $\{24,27,55,64,x\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ . | "[\"60\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2625, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16582"}, {"dataset": "skywork", "extra_inf... | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"24"
] | [
"skywork"
] | 45dcff6278789279bea0469ba457f3ae5e996a2150cd534ea94bb095c229fd0e | [
{
"content": "What is the maximum number of diagonals of a regular $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?\n\nNote: sides are not diagonals and diagonals which intersect outside the $12$ -gon at right angles are still considered perpendicular.\n\n*2018 CCA M... | What is the maximum number of diagonals of a regular $12$ -gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$ -gon at right angles are still considered perpendicular.
*2018 CCA Math Bonanza Tiebreaker Rou... | "[\"24\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2646, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16583"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"154"
] | [
"skywork"
] | 24b2e3d3e7b10817e107b580e770ee0c618be2854c43be7140a7fe9389bb67e7 | [
{
"content": "Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8.",
"role": "user"
}
] | Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. | "[\"154\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2635, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16584"}] | {
"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"
] | ea69b97603c7d4fbfc81a5e7c261c7e5487e11de18a096eae3aa4f10af4a9ffe | [
{
"content": "In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clo... | In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clocks is different, and ... | "[\"11\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2605, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "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:16585"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"11:00"
] | [
"skywork"
] | b1737c9171ab154880e7a40fece07348b04c16f58e90042510f3d523ec76de10 | [
{
"content": "One morning at 9:00 AM, a pedestrian named Fedya left the village of Fedino and headed towards the village of Novoverandovo. At the same time, a cyclist named Vera left Novoverandovo heading towards Fedya. It is known that by the time they met, Fedya had covered one-third of the distance between t... | One morning at 9:00 AM, a pedestrian named Fedya left the village of Fedino and headed towards the village of Novoverandovo. At the same time, a cyclist named Vera left Novoverandovo heading towards Fedya. It is known that by the time they met, Fedya had covered one-third of the distance between the villages. However, ... | "[\"11:00\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2612, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16586"}] | {
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[
"2"
] | [
"skywork"
] | 9742d6f6071f9594aa16addbc8f8d2b7d32e3ff20aa3d0b566dc637a1583834c | [
{
"content": "2. In the Cartesian coordinate system $x O y$, points $A$ and $B$ lie on the parabola $y^{2}=4 x$, satisfying $\\overrightarrow{O A} \\cdot \\overrightarrow{O B}=-4$, $F$ is the focus of the parabola. Then $S_{\\triangle O F A} \\cdot S_{\\triangle O F B}=$ $\\qquad$ .",
"role": "user"
}
] | 2. In the Cartesian coordinate system $x O y$, points $A$ and $B$ lie on the parabola $y^{2}=4 x$, satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B}=-4$, $F$ is the focus of the parabola. Then $S_{\triangle O F A} \cdot S_{\triangle O F B}=$ $\qquad$ . | "[\"2\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2617, "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:16587"}] | {
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[
"\\frac{\\sqrt{21}}{9}"
] | [
"skywork"
] | f149f0a07c66d9e204d805200aca0033d5da1a1fe2dc745d7c133810f6497ca0 | [
{
"content": "4. In triangle $ABC$ with angle $A$ equal to $60^{\\circ}$, the angle bisector $AD$ is drawn. The radius of the circumcircle of triangle $ADC$ with the center at point $D$ is $2 \\sqrt{3} / 3$. Find the length of segment $BM$, where $M$ is the intersection point of segments $AD$ and $BO$, if $AB=1... | 4. In triangle $ABC$ with angle $A$ equal to $60^{\circ}$, the angle bisector $AD$ is drawn. The radius of the circumcircle of triangle $ADC$ with the center at point $D$ is $2 \sqrt{3} / 3$. Find the length of segment $BM$, where $M$ is the intersection point of segments $AD$ and $BO$, if $AB=1$. | "[\"\\\\frac{\\\\sqrt{21}}{9}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2611, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 16, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16588"}] | {
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[
"\\frac{\\sqrt{3}}{3}"
] | [
"skywork"
] | 5a863694e46407dc770ac55342fb5560f3bf318df3323910a2f815d0132973b3 | [
{
"content": "On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\\frac{AM}{AC} = \\frac{CN}{CE} = \\lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\\lambda$.",
"role": "user"
}
] | On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\frac{AM}{AC} = \frac{CN}{CE} = \lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\lambda$. | "[\"\\\\frac{\\\\sqrt{3}}{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2642, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 3, "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:16589"}] | {
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[
"4"
] | [
"skywork"
] | ce2df366dcd62ad20996737e714662400bc0183d2571ef32fa4259fd21ad7990 | [
{
"content": "Let $P(x)$ be a real quadratic trinomial, so that for all $x\\in \\mathbb{R}$ the inequality $P(x^3+x)\\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.\n\n[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]",
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}
] | Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i] | "[\"4\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2641, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 11, "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:16590"}] | {
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[
"25"
] | [
"skywork"
] | cc09fd5f210d07a4ac9c1d1e036d61d4ff159fd459f31264b3e48e79db94730d | [
{
"content": "5. Let $P$ be any point on the ellipse $\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1$ other than the endpoints of the major axis, $F_{1}, F_{2}$ be its left and right foci, and $O$ be the center, then $\\left|P F_{1}\\right| \\cdot\\left|P F_{2}\\right|+|O P|^{2}=$ $\\qquad$",
"role": "user"
}
] | 5. Let $P$ be any point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ other than the endpoints of the major axis, $F_{1}, F_{2}$ be its left and right foci, and $O$ be the center, then $\left|P F_{1}\right| \cdot\left|P F_{2}\right|+|O P|^{2}=$ $\qquad$ | "[\"25\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2631, "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:16591"}] | {
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[
"x\\in(3/4;1)\\cup(1;3)"
] | [
"skywork"
] | 4d60fcfa1f58801a8e6e102d17c18eb97210669055b26358d82329a617a03ee4 | [
{
"content": "2. Solve the inequality $\\log _{x}(4 x-3)>2$.",
"role": "user"
}
] | 2. Solve the inequality $\log _{x}(4 x-3)>2$. | "[\"x\\\\in(3/4;1)\\\\cup(1;3)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2644, "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:16592"}] | {
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[
"(3;2)"
] | [
"skywork"
] | f01e1163baaaedd418db007f73163d3a2642bf0d6f2eb235dfba94c1930957f1 | [
{
"content": "7.134. $\\left\\{\\begin{array}{l}3^{2 x}-2^{y}=725 \\\\ 3^{x}-2^{y / 2}=25\\end{array}\\right.$\n\nSolve the system of equations:\n\\[\n\\left\\{\\begin{array}{l}\n3^{2 x}-2^{y}=725 \\\\\n3^{x}-2^{y / 2}=25\n\\end{array}\\right.\n\\]",
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}
] | 7.134. $\left\{\begin{array}{l}3^{2 x}-2^{y}=725 \\ 3^{x}-2^{y / 2}=25\end{array}\right.$
Solve the system of equations:
\[
\left\{\begin{array}{l}
3^{2 x}-2^{y}=725 \\
3^{x}-2^{y / 2}=25
\end{array}\right.
\] | "[\"(3;2)\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2650, "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:16593"}] | {
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[
"3(\\sqrt{3} - 1)"
] | [
"skywork"
] | 4031b6ebc75aef3ccbf65408d5d8c22b75505e9ad47427f9c537c9f8401bceac | [
{
"content": "In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin.",
"role": "user"
}
] | In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin. | "[\"3(\\\\sqrt{3} - 1)\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2647, "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:16594"}] | {
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[
"17"
] | [
"skywork"
] | 0734f91cad90910e7654d12cb12f5c573f921f4f33ae694be8a78629eb49a41d | [
{
"content": "A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\\frac{1}{2017}$ and $\\frac{1}{2016}$ unit... | A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum a... | "[\" 17 \"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2640, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 15, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16595"}] | {
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[
"1"
] | [
"skywork"
] | 735250c9a4e388834f1a5b09fc22387ce7565f2af589ec90fd2548c7b8f91c6f | [
{
"content": "13.269. At the pier, two passengers disembarked from a steamship and headed to the same village. One of them walked the first half of the way at a speed of 5 km/h and the second half at a speed of 4 km/h. The other walked the first half of the time at a speed of 5 km/h and the second half at a spe... | 13.269. At the pier, two passengers disembarked from a steamship and headed to the same village. One of them walked the first half of the way at a speed of 5 km/h and the second half at a speed of 4 km/h. The other walked the first half of the time at a speed of 5 km/h and the second half at a speed of 4 km/h, arriving... | "[\"1\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2637, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 15, "DeepSeek-R1-Distill-Qwen-7B": 14}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16596"}] | {
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[
"57.98"
] | [
"skywork"
] | 3ef3a1565eff57bad7a48b25bcd49c004eaa259fdf057f61d0ce474f77a57d33 | [
{
"content": "Find the largest solution to the equation \\[\\lfloor x \\rfloor = 8 + 50 \\{ x \\},\\] where $\\{x\\} = x - \\lfloor x \\rfloor.$",
"role": "user"
}
] | Find the largest solution to the equation \[\lfloor x \rfloor = 8 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$ | "[\"57.98\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2660, "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:16597"}] | {
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"reference_components": 1,
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} |
[
"240"
] | [
"skywork"
] | 278a8dd4568935b7d81a4a50111e4f8cbe3efb4195ad96de73fea5796768a725 | [
{
"content": "A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run at least to catch u... | A rabbit escapes and runs 100 steps ahead before a dog starts chasing it. The rabbit can cover 8 steps in the same distance that the dog can cover in 3 steps. Additionally, the dog can run 4 steps in the same time that the rabbit can run 9 steps. How many steps must the dog run at least to catch up with the rabbit? | "[\"240\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2638, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "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:16598"}] | {
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} |
[
"11,15,74"
] | [
"skywork"
] | 2f847bcbe6081c900f6899b12176669e5a3a4c7fce6ee6aae5535ff8b0323d09 | [
{
"content": "212. Divide one hundred measures of wheat among one hundred persons so that each man receives 3, each woman 2, and each child $\\frac{1}{2}$ measure. How many men, women, and children?",
"role": "user"
}
] | 212. Divide one hundred measures of wheat among one hundred persons so that each man receives 3, each woman 2, and each child $\frac{1}{2}$ measure. How many men, women, and children? | "[\"11,15,74\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2648, "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:16599"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"893"
] | [
"skywork"
] | 9b172826493cad5bbcbe01d212dc2e126ec86e5a82e9e021d90be2701c1a394e | [
{
"content": "The lengths of the sides of a triangle with positive area are $\\log_{10} 12$ , $\\log_{10} 75$ , and $\\log_{10} n$ , where $n$ is a positive integer. Find the number of possible values for $n$ .",
"role": "user"
}
] | The lengths of the sides of a triangle with positive area are $\log_{10} 12$ , $\log_{10} 75$ , and $\log_{10} n$ , where $n$ is a positive integer. Find the number of possible values for $n$ . | "[\"893\"]" | train-math-aime19832023 | [{"dataset": "skywork", "extra_info": {"index": 2657, "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:16600"}, {"dataset": "skywork", "extra_inf... | {
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} |
[
"-\\frac{1}{2}"
] | [
"skywork"
] | 6c1763e55b536ac43b866e3104c7e132062bd447ffb9f530636dadade6778218 | [
{
"content": "Let \\( n = 1990 \\), then evaluate the expression \\(\\frac{1}{2^{n}}\\left(1 - 3 C_{n}^{2} + 3^{2} C_{n}^{4} - 3^{3} C_{n}^{6} + \\cdots + 3^{994} C_{n}^{1988} - 3^{995} C_{n}^{1990}\\right)\\).",
"role": "user"
}
] | Let \( n = 1990 \), then evaluate the expression \(\frac{1}{2^{n}}\left(1 - 3 C_{n}^{2} + 3^{2} C_{n}^{4} - 3^{3} C_{n}^{6} + \cdots + 3^{994} C_{n}^{1988} - 3^{995} C_{n}^{1990}\right)\). | "[\"-\\\\frac{1}{2}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2645, "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:16601"}] | {
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} |
[
"-\\frac{p}{6}"
] | [
"skywork"
] | 9c7a56af1058e6688a6330e7f80baad78b11ac6d9cf2c41b2822369e2964a5b4 | [
{
"content": "\\left[\\begin{array}{l}\\text { Integral and Area } \\\\ \\text { [Quadratic Equations. Vieta's Theorem] }\\end{array}\\right]\n\nAuthor: Sergiev I.N.\n\nThe numbers $p$ and $q$ are such that the parabolas $y=-2 x^{2}$ and $y=x^{2}+p x+q$ intersect at two points, enclosing a certain figure.\n\nFi... | \left[\begin{array}{l}\text { Integral and Area } \\ \text { [Quadratic Equations. Vieta's Theorem] }\end{array}\right]
Author: Sergiev I.N.
The numbers $p$ and $q$ are such that the parabolas $y=-2 x^{2}$ and $y=x^{2}+p x+q$ intersect at two points, enclosing a certain figure.
Find the equation of the vertical line... | "[\"-\\\\frac{p}{6}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2636, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 2, "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:16602"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
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} |
[
"9999"
] | [
"skywork"
] | a279cb200cd236b82ca47c7a2d87a4df0abeb339cdb2e80beab5a626d4e21d51 | [
{
"content": "Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)",
"role": "user"
}
] | Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.) | "[\"9999\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2629, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 8, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16603"}, {"dataset": "skywork", "extra_in... | {
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[
"\\frac{8}{35}"
] | [
"skywork"
] | 5e32c3001255278bf38224b1a268e819f46b305c7d69c7a3a12d776d61e6a35b | [
{
"content": "A bag contains two red, two blue, two green, and one yellow marble. Three marbles are randomly selected without replacement. What is the probability that one marble of each selected is red, blue, and green (excluding the yellow marble)?",
"role": "user"
}
] | A bag contains two red, two blue, two green, and one yellow marble. Three marbles are randomly selected without replacement. What is the probability that one marble of each selected is red, blue, and green (excluding the yellow marble)? | "[\"\\\\frac{8}{35}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 2668, "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:16604"}] | {
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[
"2547"
] | [
"skywork"
] | 8fd20d101fd2383fcb3deb31d36485e1a49e086d5c0583a50e23f7f4891f6245 | [
{
"content": "Let $a_n$ and $b_n$ to be two sequences defined as below:\n\n$i)$ $a_1 = 1$\n\n$ii)$ $a_n + b_n = 6n - 1$\n\n$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n$.\n\nDetermine $a_{2009}$.",
"role": "user"
}
] | Let $a_n$ and $b_n$ to be two sequences defined as below:
$i)$ $a_1 = 1$
$ii)$ $a_n + b_n = 6n - 1$
$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.
Determine $a_{2009}$. | "[\"2547\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2651, "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:16605"}] | {
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[
"1210"
] | [
"skywork"
] | 043b86b83ca710889d4c0f73129a2940bd6481ed93d6e18786e1a0c5987c4ec2 | [
{
"content": "One evening a theater sold 300 tickets for a concert. Each ticket sold for \\$40, and all tickets were purchased using \\$5, \\$10, and \\$20 bills. At the end of the evening the theater had received twice as many \\$10 bills as \\$20 bills, and 20 more \\$5 bills than \\$10 bills. How many bills ... | One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altoge... | "[\"1210\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2639, "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:16606"}] | {
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[
"1/2"
] | [
"skywork"
] | a15679d70e6c90764783334fd12951434dee4b9fb6c858d777fe8521ae09b953 | [
{
"content": "Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squa... | Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the ... | "[\"1/2\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2649, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16607"}] | {
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[
"x=\\left(1-\\frac{\\sqrt{2}}{2}\\right) a"
] | [
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] | 1b0ed3e89297f00f09e07a84fa4495615ae2a259b732dafe6bc7fda51ede64ce | [
{
"content": "Example 1. Place four equal small balls on a plane so that the line connecting their centers forms a square, with each side having two balls touching, and place one more equal ball on top of these four balls so that it touches all of them. Given the highest point of the top ball is a. Find the rad... | Example 1. Place four equal small balls on a plane so that the line connecting their centers forms a square, with each side having two balls touching, and place one more equal ball on top of these four balls so that it touches all of them. Given the highest point of the top ball is a. Find the radius $x$ of the small b... | "[\"x=\\\\left(1-\\\\frac{\\\\sqrt{2}}{2}\\\\right) a\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2652, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16608"}] | {
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[
"5"
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"skywork"
] | 471732f03c9a7157fcd7f3e9a4433603fa9ab7a5a771f606b2549fc48db95881 | [
{
"content": "Example 17 As shown in Figure 1.4.19, a semicircle is constructed inside trapezoid $ABCD$, tangent to the upper base and the two non-parallel sides, with its diameter lying on the lower base. If $AB=2, CD=3$, find the length of the lower base $BC$.\n\n untranslated text remains the same as the ori... | Example 17 As shown in Figure 1.4.19, a semicircle is constructed inside trapezoid $ABCD$, tangent to the upper base and the two non-parallel sides, with its diameter lying on the lower base. If $AB=2, CD=3$, find the length of the lower base $BC$.
untranslated text remains the same as the original, only the content ... | "[\"5\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2655, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 13, "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:16609"}] | {
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[
"219"
] | [
"skywork"
] | 9b1208b18b4c185047201d416bd36539678d090672c6041fdfd69444729ea5ae | [
{
"content": "Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.",
"role": "user"
}
] | Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$. | "[\"219\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2663, "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:16610"}] | {
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[
"136"
] | [
"skywork"
] | 3133b39be582e7e80d3ee9212fe14a86db28392dd4e0ddf5b82f8b9b16916c14 | [
{
"content": "Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming ... | Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique? | "[\"136\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2659, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 10, "DeepSeek-R1-Distill-Qwen-32B": 12, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16611"}] | {
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[
"\\frac{9}{25}"
] | [
"skywork"
] | d2918551bc275b015881d76f87a57888670f06724c6d56d7431e447cdd5a8ced | [
{
"content": "Two mathematicians were both born in the last 600 years. Each lives (or will live) to be 120 years old, then dies. Each mathematician is equally likely to be born at any point during those 600 years. What is the probability that they were contemporaries for any length of time?",
"role": "user"... | Two mathematicians were both born in the last 600 years. Each lives (or will live) to be 120 years old, then dies. Each mathematician is equally likely to be born at any point during those 600 years. What is the probability that they were contemporaries for any length of time? | "[\"\\\\frac{9}{25}\"]" | train-math-still3 | [{"dataset": "skywork", "extra_info": {"index": 2674, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 1, "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:16612"}] | {
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[
"\\frac{\\sqrt{3}}{3}"
] | [
"skywork"
] | 95796ff32d86769b75be0f7d6dfab41f28503fc35cd2d4273084e50661bcf47f | [
{
"content": "The centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$.",
"role": "user"
}
] | The centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$. | "[\"\\\\frac{\\\\sqrt{3}}{3}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2662, "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:16613"}] | {
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[
"116"
] | [
"skywork"
] | d8de0580e81b7353664aede1b8e867553fa5fbca164a607ec8aa75d936f8d83e | [
{
"content": "7. Xiao Le has some balls, and he puts them into 25 small squares, with at least 1 ball in each small square. No matter how he places them, there are always 4 small squares with the same number of balls. Xiao Le can have at most $\\qquad$ balls.",
"role": "user"
}
] | 7. Xiao Le has some balls, and he puts them into 25 small squares, with at least 1 ball in each small square. No matter how he places them, there are always 4 small squares with the same number of balls. Xiao Le can have at most $\qquad$ balls. | "[\"116\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2656, "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:16614"}] | {
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[
"120"
] | [
"skywork"
] | 85de81c4dbcd4791640610e76d465825d6f300958b817aa6b39493c5dd800a45 | [
{
"content": "Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods.",
"role": "user"
}
] | Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods. | "[\"120\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2666, "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:16615"}] | {
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} |
[
"\\[\n\\boxed{y = -\\frac{1}{4}}\n\\]"
] | [
"skywork"
] | 1f4057c8d8c34784d3bbef509b87f3dad5c1e19beb4a90d43de9566961320371 | [
{
"content": "For a point $P = (a, a^2)$ in the coordinate plane, let $\\ell(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$ , $P_2 = (a_2, a_2^2)$ , $P_3 = (a_3, a_3^2)$ , such that the intersections of the lines $\\ell(P_1... | For a point $P = (a, a^2)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$ , $P_2 = (a_2, a_2^2)$ , $P_3 = (a_3, a_3^2)$ , such that the intersections of the lines $\ell(P_1)$ , $\ell(P_2)$ , $\ell... | "[\"\\\\[\\n\\\\boxed{y = -\\\\frac{1}{4}}\\n\\\\]\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2643, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 9, "DeepSeek-R1-Distill-Qwen-7B": 12}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16616"}] | {
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[
"17"
] | [
"skywork"
] | c0e39fbd91f9656157e9ab22ff6d706ca77ba871dca8a53ba520e886e21912c0 | [
{
"content": "Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.\n\n$\\begin{array}{@{\\hspace{-1em}}l@{\\hspace{14em}}l@{\\hspace{14em}}l} \n\n\\textbf{(A) }0&\\textbf{(B) }1&\\textbf{(C) }2\\\\\\\\ ... | Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\text... | "[\"17\"]" | train-math-numinamath1.5_aops_forum | [{"dataset": "skywork", "extra_info": {"index": 2658, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 14, "DeepSeek-R1-Distill-Qwen-7B": 16}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16617"}] | {
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[
"137"
] | [
"skywork"
] | 8a3080e95dc2c5e2e89af3b1d9b0e8a17bf6435734472c1b8b684a05ce7ace9c | [
{
"content": "Given a tetrahedron \\( A B C D \\) with side lengths \\( A B = 41 \\), \\( A C = 7 \\), \\( A D = 18 \\), \\( B C = 36 \\), \\( B D = 27 \\), and \\( C D = 13 \\), let \\( d \\) be the distance between the midpoints of edges \\( A B \\) and \\( C D \\). Find the value of \\( d^{2} \\).",
"rol... | Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \). | "[\"137\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2661, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 8}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16618"}] | {
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[
"351"
] | [
"skywork"
] | 6931aa344d834a3f75e80168aa72a2908083267c4b66abe9a5df07fe9dbdfe61 | [
{
"content": "1. How many non-empty subsets of the set $\\{0,1, \\ldots, 9\\}$ have the sum of their elements divisible by three?\n\n(Eliška Macáková)",
"role": "user"
}
] | 1. How many non-empty subsets of the set $\{0,1, \ldots, 9\}$ have the sum of their elements divisible by three?
(Eliška Macáková) | "[\"351\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2683, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 9, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16619"}] | {
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[
"16+2\\sqrt{13}"
] | [
"skywork"
] | bad88e041e79a9957d00d837262f4522bf9da452b791a7029a918b97cfa0b5f7 | [
{
"content": "8. In a right trapezoid, the upper base is shorter than the lower base. If the volume of the solid of revolution obtained by rotating it around its lower base is $80 \\pi$,\n\nthe volume of the solid of revolution obtained by rotating it around its upper base is $112 \\pi$, and the volume of the s... | 8. In a right trapezoid, the upper base is shorter than the lower base. If the volume of the solid of revolution obtained by rotating it around its lower base is $80 \pi$,
the volume of the solid of revolution obtained by rotating it around its upper base is $112 \pi$, and the volume of the solid of revolution obtaine... | "[\"16+2\\\\sqrt{13}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2653, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 16, "DeepSeek-R1-Distill-Qwen-32B": 6, "DeepSeek-R1-Distill-Qwen-7B": 13}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16620"}] | {
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} |
[
"7"
] | [
"skywork"
] | 063830aa9920714aa4965040bc498e09f6ed19a4875bbeaea67950fbadb54e9f | [
{
"content": "Exercise 5. Martin is trying to fill each cell of a rectangular grid with 8 rows and $\\mathrm{n}$ columns with one of the four letters $\\mathrm{P}, \\mathrm{O}, \\mathrm{F}$, and $\\mathrm{M}$ such that for any pair of distinct rows, there is at most one column where the intersections of the two... | Exercise 5. Martin is trying to fill each cell of a rectangular grid with 8 rows and $\mathrm{n}$ columns with one of the four letters $\mathrm{P}, \mathrm{O}, \mathrm{F}$, and $\mathrm{M}$ such that for any pair of distinct rows, there is at most one column where the intersections of the two rows are cells with the sa... | "[\"7\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2673, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 8, "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:16621"}] | {
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} |
[
"2x+1"
] | [
"skywork"
] | 78696818d385522ce0290ab2a864e0a7d0f270a0f7078ef0c20040080301c5d6 | [
{
"content": "1. $x^{5}+x+1$ when divided by $x^{2}+1$ yields a remainder of",
"role": "user"
}
] | 1. $x^{5}+x+1$ when divided by $x^{2}+1$ yields a remainder of | "[\"2x+1\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2678, "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:16622"}] | {
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} |
[
"18"
] | [
"skywork"
] | 8ab7ad2d0d35c02b60e8c127ee4665e31d94413b7519aa6dc3c27ee9966be8f8 | [
{
"content": "9.5 In some cells of a $10 \\times 10$ table, crosses are placed such that each of them is the only one either in its row or in its column. What is the maximum number of crosses that can be in such a table? Justify your answer.",
"role": "user"
}
] | 9.5 In some cells of a $10 \times 10$ table, crosses are placed such that each of them is the only one either in its row or in its column. What is the maximum number of crosses that can be in such a table? Justify your answer. | "[\"18\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2669, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 15, "DeepSeek-R1-Distill-Qwen-32B": 15, "DeepSeek-R1-Distill-Qwen-7B": 15}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16623"}] | {
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[
"A"
] | [
"skywork"
] | 3dfd837a1a1170e8acc98f084e0724628ff72687ae4a8593083bb075ccd52f05 | [
{
"content": "6. If $F\\left(\\frac{1-x}{1+x}\\right)=x$, then which of the following equations is correct?\nA. $F(-2-x)=-2-F(x)$\nB. $F(-x)=F\\left(\\frac{1+x}{1-x}\\right)$\nC. $F\\left(x^{-1}\\right)=F(x)$\nD. $F(F(x))=-x$",
"role": "user"
}
] | 6. If $F\left(\frac{1-x}{1+x}\right)=x$, then which of the following equations is correct?
A. $F(-2-x)=-2-F(x)$
B. $F(-x)=F\left(\frac{1+x}{1-x}\right)$
C. $F\left(x^{-1}\right)=F(x)$
D. $F(F(x))=-x$ | "[\"A\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2672, "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:16624"}] | {
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"plain_numeric_reference": false,
"reference_components": 1,
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[
"114"
] | [
"skywork"
] | 27531d796a0c0e3c24298923752c553f46cbf4ca0274d56bdb720695e8ad545b | [
{
"content": "3. From the set $\\{1,2,3, \\cdots, 20\\}$, any 4 different numbers are taken such that these 4 numbers form an arithmetic sequence. The number of such arithmetic sequences is\nA. 110\nB. 114\nC. 116\nD. 120",
"role": "user"
}
] | 3. From the set $\{1,2,3, \cdots, 20\}$, any 4 different numbers are taken such that these 4 numbers form an arithmetic sequence. The number of such arithmetic sequences is
A. 110
B. 114
C. 116
D. 120 | "[\"114\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2679, "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:16625"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{\\pi}{3};\\frac{2\\sqrt{3}}{3}"
] | [
"skywork"
] | 11b0365b5da10413210dd5f3a28a1a8baa49f6ee908687d89dc830a23e353c9d | [
{
"content": "11.231 The base of the pyramid $S A B C$ is a right isosceles triangle $A B C$, with the length of the hypotenuse $A B=4 \\sqrt{2}$. The lateral edge $S C$ of the pyramid is perpendicular to the plane of the base, and its length is 2. Find the measure of the angle and the distance between the line... | 11.231 The base of the pyramid $S A B C$ is a right isosceles triangle $A B C$, with the length of the hypotenuse $A B=4 \sqrt{2}$. The lateral edge $S C$ of the pyramid is perpendicular to the plane of the base, and its length is 2. Find the measure of the angle and the distance between the lines, one of which passes ... | "[\"\\\\frac{\\\\pi}{3};\\\\frac{2\\\\sqrt{3}}{3}\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2665, "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:16626"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"900"
] | [
"skywork"
] | 7c00a0d86af21de17323cd11dfdb7541c17ec16e5de46a6ce6bdaea14413fb80 | [
{
"content": "João and Maria each have a large jug with one liter of water. On the first day, João puts $1 \\mathrm{ml}$ of water from his jug into Maria's jug. On the second day, Maria puts $2 \\mathrm{ml}$ of water from her jug into João's jug. On the third day, João puts $3 \\mathrm{ml}$ of water from his ju... | João and Maria each have a large jug with one liter of water. On the first day, João puts $1 \mathrm{ml}$ of water from his jug into Maria's jug. On the second day, Maria puts $2 \mathrm{ml}$ of water from her jug into João's jug. On the third day, João puts $3 \mathrm{ml}$ of water from his jug into Maria's jug, and s... | "[\"900\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2671, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 5, "DeepSeek-R1-Distill-Qwen-32B": 0, "DeepSeek-R1-Distill-Qwen-7B": 0}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16627"}] | {
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"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"225"
] | [
"skywork"
] | 631efa178947e39a9148a8515362352fd88e04ab5bccded417781c4e14b3f4c6 | [
{
"content": "We consider a white \\( 5 \\times 5 \\) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle?",
"role": "user"
}
] | We consider a white \( 5 \times 5 \) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle? | "[\"225\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2677, "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:16628"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"1/3 + 1/(3 \\cdot 2^{2003})"
] | [
"skywork"
] | 18089dc0697fd4b3716a1ce7e8a2147b976f869af43192bc5553ba88e6e448a6 | [
{
"content": "In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches... | In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with ... | "[\"1/3 + 1/(3 \\\\cdot 2^{2003})\"]" | train-math-omnimath | [{"dataset": "skywork", "extra_info": {"index": 2675, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 14, "DeepSeek-R1-Distill-Qwen-32B": 1, "DeepSeek-R1-Distill-Qwen-7B": 6}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16629"}] | {
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"label_correctness_independently_verified": false,
"plain_numeric_reference": false,
"reference_components": 1,
"reference_present": true
} |
[
"2x-y+7z-23=0"
] | [
"skywork"
] | b6a0cb536c604d713b12dc5977ac7d66753a6198bf80203b392ec1d966cc1b53 | [
{
"content": "[ $\\underline{\\text { equation of a plane ] }}$ [ Scalar product ]\n\nForm the equation of the plane containing the line $\\frac{x-1}{2}=-\\frac{y}{3}=3-z$ and parallel to the line of intersection of the planes $4 x+5 z-3=0$ and $2 x+y+2 z=0$.",
"role": "user"
}
] | [ $\underline{\text { equation of a plane ] }}$ [ Scalar product ]
Form the equation of the plane containing the line $\frac{x-1}{2}=-\frac{y}{3}=3-z$ and parallel to the line of intersection of the planes $4 x+5 z-3=0$ and $2 x+y+2 z=0$. | "[\"2x-y+7z-23=0\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2670, "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:16630"}] | {
"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
} |
[
"480"
] | [
"skywork"
] | 0287080c856e9eca0172f0cba2e81748a71b3267f1e12f34c3ec02309cc05cdf | [
{
"content": "6. (3 points) Person A and Person B work together to process a batch of parts, which can be completed in 8 hours. If Person A works alone, it would take 12 hours to complete the task. Now, Person A and Person B work together for $2 \\frac{2}{5}$ hours, after which Person A is reassigned to other w... | 6. (3 points) Person A and Person B work together to process a batch of parts, which can be completed in 8 hours. If Person A works alone, it would take 12 hours to complete the task. Now, Person A and Person B work together for $2 \frac{2}{5}$ hours, after which Person A is reassigned to other work, and Person B conti... | "[\"480\"]" | train-math-numinamath1.5_olympiads | [{"dataset": "skywork", "extra_info": {"index": 2667, "model_difficulty": {"DeepSeek-R1-Distill-Qwen-1.5B": 12, "DeepSeek-R1-Distill-Qwen-32B": 2, "DeepSeek-R1-Distill-Qwen-7B": 4}}, "file": "data/math-00000-of-00001.parquet", "row_id": "skywork:data/math-00000-of-00001.parquet:16631"}] | {
"all_reference_components_parse": true,
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"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
[
"\\frac{83}{499}"
] | [
"skywork"
] | 815a3f17da353cafaa19a4851c5dc7a679c3d9caa3c36a0157565f2d5d4b98b1 | [
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Find the probability that\n\\[\\sqrt{2+\\sqrt{3}}\\le\\left|v+w\\right|.\\]",
"role": "user"
}
] | Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Find the probability that
\[\sqrt{2+\sqrt{3}}\le\left|v+w\right|.\] | "[\"\\\\frac{83}{499}\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2685, "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:16632"}] | {
"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
} |
[
"6"
] | [
"skywork"
] | f09fe56f643a190c776b6b83ca8f51d4b90db438d702e38a390e050b902bd1b8 | [
{
"content": "How many positive integers $n$ satisfy\\[\\dfrac{n+1000}{70} = \\lfloor \\sqrt{n} \\rfloor?\\](Recall that $\\lfloor x\\rfloor$ is the greatest integer not exceeding $x$.)\n$\\textbf{(A) } 2 \\qquad\\textbf{(B) } 4 \\qquad\\textbf{(C) } 6 \\qquad\\textbf{(D) } 30 \\qquad\\textbf{(E) } 32$",
"r... | How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$ | "[\"6\"]" | train-math-deepscaler | [{"dataset": "skywork", "extra_info": {"index": 2684, "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:16633"}] | {
"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"
] | 351300fedf0c220520db4f26dfa05b616fb1f0a4ebe91f2a8bb264482ba3d94f | [
{
"content": "Example 2. If $x^{3}-x^{2}+x-2=0$. Then $x^{4}+2 x^{3}-2 x^{2}+x-1=$ $\\qquad$ . (1991, Hubei Huanggang Region Mathematics Competition).",
"role": "user"
}
] | Example 2. If $x^{3}-x^{2}+x-2=0$. Then $x^{4}+2 x^{3}-2 x^{2}+x-1=$ $\qquad$ . (1991, Hubei Huanggang Region Mathematics Competition). | "[\"5\"]" | train-math-numinamath1.5_cn_contest | [{"dataset": "skywork", "extra_info": {"index": 2694, "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:16634"}] | {
"all_reference_components_parse": true,
"any_reference_component_parses": true,
"label_correctness_independently_verified": false,
"plain_numeric_reference": true,
"reference_components": 1,
"reference_present": true
} |
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