messages listlengths 2 2 | num_tokens int64 82 19k | json_metadata stringclasses 1
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[
{
"content": "In triangle $ABC$ with $BC=3$, $CA=4$, $AB=5$ and $\\angle C = 90^{\\circ}$, there exists exactly one circle $\\omega$ that is tangent to segment $AB$ and rays $\\overrightarrow{CB}$ and $\\overrightarrow{CA}$. Let $A'$ be the image of the reflection of point $A$ over the line $BC$. Compute the sq... | 2,027 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Let \\(a, b, c > 0\\) satisfy the conditions:\n\\[ c \\leq b < a \\leq 3, \\]\n\\[ b^2 + 2a \\leq 10, \\]\n\\[ b^2 + 2a + 2c \\leq 14, \\]\n\\[ (a^2 + 1)(b^2 + 1) + 4ab \\leq 2a^3 + 2b^3 + 2a + 2b. \\]\nFind the maximum value of \\(P = 4a^4 + b^4 + 2b^2 + 4c^2\\).",
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{
... | 3,523 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "A car drives up a hill. It reaches point $A$ and then passes 30 m in the first second and 2 m less than during the preceding second for every subsequent second. Nine seconds after the car reached point $A$, a bus starts from point $B$, which is 258 m from $A$, and drives towards the car. In the f... | 2,353 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Let XY be a diameter of a circle with radius 10 centered at O. Points A and B are on XY such that X, A, O, B, Y are in this order and AO = OB = 4. Suppose P is a point on the circle such that the lines PA and PB intersect the circle again at points C and D, respectively. Given that PB/BD = 16/21, ... | 2,131 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Given the sequence defined by \\(a_1 = \\sqrt{2}\\), \\(a_2 = \\sqrt{3}\\), and \\(a_n = a_{n-1} + 2 \\lfloor a_{n-2} \\rfloor\\) for \\(n \\geq 3\\), find the last two digits of \\(\\lfloor a_{2021} \\rfloor\\).",
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{
"content": "<think>We need to compute floor of a_n ... | 3,515 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "In the triangular pyramid $SABC$ the edge $SA$ is perpendicular to the plane of the face $ABC$, the dihedral angle with the edge $SC$ is $\\pi /4$, $| SA | = | BC | = a$ and $\\angle ABC$ is a right angle. Find the length of the edge $A B$.",
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{
"content": "<think>We h... | 3,884 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "A rectangular sheet of paper is folded so that the two diagonally opposite corners come together. If the crease formed is of the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?",
"role": "user"
},
{
"content": "<think>... | 2,746 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Calculate the derivatives of the following functions. $f(x)=\\sqrt {x+2\\sqrt {2x-4}}+\\sqrt {x-2\\sqrt {2x-4}}$.",
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{
"content": "<think>We need derivative of f(x) = sqrt(x + 2 sqrt(2x-4)) + sqrt(x - 2 sqrt(2x-4)). Domain: need 2x-4 >=0 => x>=2. Also inside sqrt: x ± ... | 2,282 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty} n \\ln\\left(1+\\ln\\left(1+\\ln\\left(1+ \\cdots + \\ln\\left(1+\\frac{2007}{n}\\right)\\right)\\right)\\right) \\), where the number of natural logarithms is equal to \\( n \\).",
"role": "user"
},
{
"content": "<think>We need to evaluate l... | 2,530 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Given that \\( f(0) = \\frac{2}{3} \\) and for all \\( n = 1, 2, \\ldots \\), \\( f(n) \\neq 0 \\), and the recurrence relation \\((f(n+1) - 1)(f(n) + 3) + 3 = 0\\), find the value of \\(\\frac{1}{f(0)} + \\frac{1}{f(1)} + \\frac{1}{f(2)} + \\cdots + \\frac{1}{f(2016)} + \\frac{1}{f(2017)}\\).",
... | 1,998 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "In $\\Delta ABC$, $AB=5$ and points $E$ and $F$ are on $BC$ such that $BE=1$, $EF=3$, and $CF=2$. The lines $AE$ and $AF$ intersect the circumcircle of $\\Delta ABC$ at points $G$ and $H$ respectively. Given that $GH$ is parallel to $BC$, find the length of $AC$.",
"role": "user"
},
{
... | 5,384 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Find an analytic function \\( f: G \\to \\mathbb{C} \\) where \\( G = \\{ z \\ | \\ \\text{Re}(z) > 0 \\} \\) such that \\( f(G) = D = \\{ z \\ | \\ |z| < 1 \\} \\).",
"role": "user"
},
{
"content": "<think>We need to find a conformal map from right half-plane to unit disk. Classic map... | 991 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Determine the five smallest natural numbers that have exactly 18 positive factors.",
"role": "user"
},
{
"content": "<think>We need smallest natural numbers (positive integers) that have exactly 18 positive divisors. Find the five smallest such numbers.\n\nWe need to recall divisor fun... | 5,562 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose the chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. What is the number of possible arrangements?",
"role": "user"
},
... | 876 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Calculate the distance between the lines $BF'$ and $ND$ in the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with base edge length $12$ and height $12 \\sqrt{3}$, where $N$ is the midpoint of the edge $CC'$.",
"role": "user"
},
{
"content": "<think>We have a regular hexagonal prism. Bas... | 3,473 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Find the number of rectangles whose area, length, and breadth are digits (excluding zero). Also, find the dimensions of the rectangle with the greatest area.",
"role": "user"
},
{
"content": "<think>We need to interpret the problem: \"Find the number of rectangles whose area, length, a... | 2,282 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Find the volume of a regular octahedron all of whose vertices are on a sphere of diameter two.",
"role": "user"
},
{
"content": "<think>We need volume of regular octahedron inscribed in a sphere of diameter 2, i.e., sphere radius R = 1. Regular octahedron: 8 faces, each an equilateral ... | 972 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Find the smallest positive integer $ n$ such that $ 2014^n - n^{2014}$ is divisible by $ 11$.",
"role": "user"
},
{
"content": "<think>We need smallest positive integer n such that 2014^n - n^{2014} ≡ 0 mod 11.\n\nCompute modulo 11. First reduce 2014 mod 11. 11*183 = 2013, so 2014 ≡ 1 ... | 1,597 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Find the volume formed by the revolution of the disk enclosed by the circle passing through the points \\( A(1,0,0) \\), \\( B(0,1,0) \\), and \\( C(0,0,1) \\) about the \\( x \\)-axis.",
"role": "user"
},
{
"content": "<think>We need to find the volume formed by revolving the disk (i.... | 8,156 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Evaluate the limit $\\lim_{x\\to 0} \\frac{e^{x^2}-\\cos x}{x^2}$.",
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},
{
"content": "<think>We need to compute limit as x→0 of (e^{x^2} - cos x) / x^2.\n\nWe can expand series.\n\ne^{x^2} = 1 + x^2 + x^4/2 + x^6/6 + ...\n\ncos x = 1 - x^2/2 + x^4/24 - x^6/720 + ...\n\nS... | 1,265 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Two factories, $A$ and $B$, undertook to fulfil an order in 12 days. After two days factory $A$ was closed down for repairs, while factory $B$ continued fulfilment of the order. Knowing that $B$ has an efficiency of $66\\frac{2}{3}\\%$ of that of $A$, determine in how many days the order will be c... | 1,078 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Let \\( f : x \\rightarrow x^2 + 1 \\). Calculate \\( (f^{2022})^{(100)}(1) \\mod (2^{89} - 1) \\).\n\n Here, \\( f^2 = f \\circ f \\) and \\( f^{(2)} = f'' \\).",
"role": "user"
},
{
"content": "<think>We need to parse the problem: f: x -> x^2 + 1. Compute (f^{2022})^{(100)}(1) mod (... | 5,932 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
[
{
"content": "Solve the equation $3^x \\times 8^{\\frac{x}{x+2}} = 6$. If the sum of all real solutions can be expressed in the form $-a - b \\log_c d$, where $a, b, c, d$ are positive integers and $b > 1$, find the minimum value of $a + 2b + 3c + 4d$.",
"role": "user"
},
{
"content": "<think>We nee... | 2,871 | {"source": "Jackrong/gpt-oss-120B-distilled-reasoning", "license": "apache-2.0"} |
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