data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven an infinite number of non-degenerate triangles, sum their shortest sides, second longest sides, and longest sides to form a new triangle. What is the minimum value of the cosine of the largest angle of this new triangle?\n\nPlease reason step by step, and put yo... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93821,
"sample_id": "deepmath-103k:093821",
"difficulty": 8,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 74,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_res... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of real solutions to the equation \\(2^{\\sin x} + 2^{\\cos x} = 2^{3x^2 - 2x^3 - 1}\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 18027,
"sample_id": "deepmath-103k:018027",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 73,
"p0_stage": "truncation-rescued-train",
"gap_label": "tru... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet \\( x, y \\) be nonnegative integers such that \\( x + 2y \\) is a multiple of 5, \\( x + y \\) is a multiple of 3, and \\( 2x + y \\geq 99 \\). Find the minimum possible value of \\( 7x + 5y \\).\n\nPlease reason step by step, and put your final answer within \\b... | math | {
"style": "rule",
"ground_truth": "366"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 91753,
"sample_id": "deepmath-103k:091753",
"difficulty": 5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 101,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_stric... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet \\( \\omega \\) be a complex number such that \\( \\omega^5 = 1 \\) and \\( \\omega \\neq 1 \\). Evaluate the expression:\n\\[\n\\frac{\\omega}{1 - \\omega^2} + \\frac{\\omega^2}{1 - \\omega^4} + \\frac{\\omega^3}{1 - \\omega} + \\frac{\\omega^4}{1 - \\omega^3}\n\... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 11077,
"sample_id": "deepmath-103k:011077",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"topic_family": "algebra",
"prompt_tokens": 127,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncati... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nCalculate the value of the sum \\(S=\\sum_{i=1}^{4}\\frac{1}{\\alpha_i^4-15\\alpha_i^2-10\\alpha_i+24}\\), where \\(\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4\\) are the roots of the polynomial \\(p(x)=x^4+4x^3+x^2-6x-1=0\\), expressed in the form \\(\\frac{m}{n}\\) with... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 38490,
"sample_id": "deepmath-103k:038490",
"difficulty": 7.5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 173,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_re... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the minimum number of black boxes required to pinpoint the location of the origin of a signal that caused spaceships to explode. Assume the signal spreads through space at a constant velocity uniformly in every direction, and each black box contains the spat... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 9355,
"sample_id": "deepmath-103k:009355",
"difficulty": 5.5,
"topic": "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes",
"topic_family": "geometry",
"prompt_tokens": 89,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the maximum possible value of \\(|a| + |b| + |c|\\) given that \\(|ax^2 + bx + c| \\leq 250\\) for all \\(|x| \\leq 1\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 72825,
"sample_id": "deepmath-103k:072825",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 82,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nOn a blackboard, the numbers from 1 to 2004 are written: \\(1, 2, \\cdots, 2004\\). In each step, erase some of the numbers from the blackboard and write the remainder of their sum divided by 167. After several steps, two numbers remain on the blackboard, one of which... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 88195,
"sample_id": "deepmath-103k:088195",
"difficulty": 5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 120,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_incon... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nHow many complex numbers $z$ such that $\\left| z \\right| < 30$ satisfy the equation\n\\[\n e^z = \\frac{z - 1}{z + 1} \\, ?\n\\]\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 102577,
"sample_id": "deepmath-103k:102577",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"topic_family": "algebra",
"prompt_tokens": 80,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncati... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the coefficient of \\( t^{q^2+q+1} \\) in the expansion of the polynomial \\( P(t) = \\left(\\frac{1-t^{q+1}}{1-t}\\right)^{2q-1} \\), where \\( q \\) is a positive integer. Assume \\( q \\) is a prime power and the polynomial is in the ring \\( \\mathbb{F}_q[t] ... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 37346,
"sample_id": "deepmath-103k:037346",
"difficulty": 8,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 121,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet $z_1,$ $z_2,$ $\\dots,$ $z_{20}$ be the twenty (complex) roots of the equation\n\\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \\dots + 441 = 0.\\]Calculate $\\cot \\left( \\sum_{k = 1}^{20} \\operatorname{arccot} z_k \\right).$ Note that the addition formula for cot... | math | {
"style": "rule",
"ground_truth": "\\frac{241}{220}"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 95111,
"sample_id": "deepmath-103k:095111",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"topic_family": "algebra",
"prompt_tokens": 154,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncati... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the smallest integer \\( n \\geq 3 \\) such that for any partition of the set \\( I = \\{1, 2, \\ldots, n\\} \\) into two non-empty disjoint subsets \\( A \\) and \\( B \\), there exist two numbers in either \\( A \\) or \\( B \\) whose sum is a perfect squa... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93797,
"sample_id": "deepmath-103k:093797",
"difficulty": 7,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 107,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind all positive integers \\( n \\) such that \\( n^{4} - n^{3} + 3n^{2} + 5 \\) is a perfect square.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 90860,
"sample_id": "deepmath-103k:090860",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 66,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncatio... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet \\( G=10^{10^{100}} \\) (a.k.a. a googolplex). Then \\[ \\log_{\\left(\\log_{\\left(\\log_{10} G\\right)} G\\right)} G \\] can be expressed in the form \\( \\frac{m}{n} \\) for relatively prime positive integers \\( m \\) and \\( n \\). Determine the sum of the di... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 89945,
"sample_id": "deepmath-103k:089945",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions",
"topic_family": "algebra",
"prompt_tokens": 127,
"p0_stage": "truncation-rescued-train",
"gap_label": "... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a class, every group of three students contains a pair of friends and a pair of enemies. Determine the maximum number of students that can be in the class.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 47175,
"sample_id": "deepmath-103k:047175",
"difficulty": 6,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 62,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a circle, points $A, B, C,$ and $D$ are arranged in order. The extensions of $AB$ and $DC$ intersect at point $E$, and the extensions of $AD$ and $BC$ intersect at point $F$. Tangents from $E$ and $F$ to the circle meet points $P$ and $Q$, respectively, with $EP = ... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 38790,
"sample_id": "deepmath-103k:038790",
"difficulty": 6,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Polygons",
"topic_family": "geometry",
"prompt_tokens": 138,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven the functional equation $f(f(n)) = 3n$ for each positive integer $n$, where $f(n+1) > f(n)$, find the value of $f(10)$.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 12632,
"sample_id": "deepmath-103k:012632",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 73,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescue... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven that the circle \\( C: x^{2}+(y-1)^{2}=r^{2} \\) and the function \\( y=\\sin x \\) have a unique intersection point, and the x-coordinate of this intersection point is \\( \\alpha \\), find the value of \\( \\frac{2 \\sin 2 \\alpha \\cos \\alpha-4 \\cos ^{2} \\... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 87147,
"sample_id": "deepmath-103k:087147",
"difficulty": 5.5,
"topic": "Mathematics -> Precalculus -> Trigonometric Functions",
"topic_family": "algebra",
"prompt_tokens": 123,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the chromatic number of the complement of the Petersen graph.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 75290,
"sample_id": "deepmath-103k:075290",
"difficulty": 5.5,
"topic": "Mathematics -> Discrete Mathematics -> Graph Theory",
"topic_family": "combinatorics",
"prompt_tokens": 44,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the smallest integer \\( n \\) such that in any \\( n \\times n \\) grid, where each cell is colored either white or black, there always exists a rectangle whose vertices are the centers of four cells of the same color.\n\nPlease reason step by step, and put... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 89614,
"sample_id": "deepmath-103k:089614",
"difficulty": 6,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 79,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the largest integer $k$ such that there exists a perfect square, not divisible by 10, whose last $k$ digits are all the same.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 43189,
"sample_id": "deepmath-103k:043189",
"difficulty": 5.5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 63,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\\triangle ABD$. Given that $\\cos^2 \\angle AEI = \\frac{m}{n}$, where $m... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 85560,
"sample_id": "deepmath-103k:085560",
"difficulty": 7,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 150,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_in... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven that the area of triangle ABC with vertices \\((a,a^2),(b,b^2),(c,c^2)\\) is \\(\\frac{1}{4}\\) square units and the area of triangle PQR with vertices \\((p,p^2),(q,q^2),(r,r^2)\\) is \\(3\\) square units, find the value of the determinant:\n\\[\n\\begin{vmatri... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 80815,
"sample_id": "deepmath-103k:080815",
"difficulty": 7,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 194,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_re... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of distinct terms in the expansion of \\(\\left(x+\\frac{1}{x}+\\frac{1}{x^2}+x^2\\right)^{15}\\) with respect to powers of \\(x\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 6539,
"sample_id": "deepmath-103k:006539",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"topic_family": "algebra",
"prompt_tokens": 81,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nPoints \\( M \\), \\( N \\), and \\( K \\) are located on the lateral edges \\( AA_1 \\), \\( BB_1 \\), and \\( CC_1 \\) of the triangular prism \\( ABC A_1 B_1 C_1 \\) such that \\( \\frac{AM}{AA_1} = \\frac{2}{3} \\), \\( \\frac{BN}{BB_1} = \\frac{3}{5} \\), \\( \\f... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 85164,
"sample_id": "deepmath-103k:085164",
"difficulty": 5.5,
"topic": "Mathematics -> Geometry -> Solid Geometry -> Volume",
"topic_family": "geometry",
"prompt_tokens": 181,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nThe largest solution to\n\\[9x^3 - 20x = 8 \\sqrt{2}\\]can be written in the form $\\frac{\\sqrt{a} + \\sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$\n\nPlease reason step by step, and put your final answer within \\b... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 100109,
"sample_id": "deepmath-103k:100109",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"topic_family": "algebra",
"prompt_tokens": 103,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncat... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of ordered triples \\((x, y, z)\\) of real numbers that satisfy the following system of equations:\n\\[ x + y^2 = z^3, \\]\n\\[ x^2 + y^3 = z^4, \\]\n\\[ x^3 + y^4 = z^5. \\]\n\nPlease reason step by step, and put your final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 33700,
"sample_id": "deepmath-103k:033700",
"difficulty": 7.5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 102,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncati... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven two graphs $G_1$ and $G_2$ with vertex chromatic numbers 5 and 7, respectively, and an edge between vertices 7 and 8 in both graphs, determine the vertex chromatic number of the graph $G$ formed by taking the union of the edge sets of $G_1$ and $G_2$. Provide yo... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 40376,
"sample_id": "deepmath-103k:040376",
"difficulty": 5,
"topic": "Mathematics -> Discrete Mathematics -> Graph Theory",
"topic_family": "combinatorics",
"prompt_tokens": 111,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of non-decreasing tuples of integers \\((a_1, a_2, \\dots, a_{16})\\) such that \\(0 \\leq a_i \\leq 16\\) for all \\(i\\), and the sum of all \\(a_i\\) is even.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300546630"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93978,
"sample_id": "deepmath-103k:093978",
"difficulty": 5.5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 93,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_res... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of integer solutions to the Diophantine equation \\(4x^3 - 3 = y^2\\) for \\(x > 0\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 40391,
"sample_id": "deepmath-103k:040391",
"difficulty": 5.5,
"topic": "Mathematics -> Number Theory -> Other",
"topic_family": "number_theory",
"prompt_tokens": 63,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inconsiste... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn an acute triangle \\(ABC\\) with \\(AB \\neq AC\\), let \\(H\\) be the foot of the altitude from \\(A\\). Points \\(P\\) and \\(Q\\) are taken on the extensions of sides \\(AB\\) and \\(AC\\) respectively, such that \\(HP = HQ\\) and the points \\(B, C, P, Q\\) are... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 89314,
"sample_id": "deepmath-103k:089314",
"difficulty": 6.5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 119,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the rank of the first homology group of the double branched cover of a knot $K$ in $S^3$.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 41149,
"sample_id": "deepmath-103k:041149",
"difficulty": 8,
"topic": "Mathematics -> Geometry -> Solid Geometry -> Other",
"topic_family": "geometry",
"prompt_tokens": 57,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nTriangle $ABC$ has sides $AB = 2$, $BC = 3$, and $CA = 4$. The circumcenter of the triangle is $O$. If the sum of the areas of triangles $AOB$, $BOC$, and $COA$ is expressed as $\\frac{a\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers with $\\gcd(a, c) = ... | math | {
"style": "rule",
"ground_truth": "152"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 90231,
"sample_id": "deepmath-103k:090231",
"difficulty": 6,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 151,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_re... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven a sequence of real numbers $a_1, a_2, \\ldots, a_{2013}$ such that $a_1 > 3$, $a_{2013} = 2013$, and for $n = 1, 2, \\ldots, 2012$, $a_{n + 1} = \\frac{-a_n + a_n\\sqrt{4a_n - 3}}{2}$, determine the value of \\[ a_1 + \\sum_{i = 1}^{2012}\\frac{a_{i + 1}^3}{a_i^... | math | {
"style": "rule",
"ground_truth": "4025"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 10111,
"sample_id": "deepmath-103k:010111",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 195,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the minimum value of the expression:\n\\[\n\\left| \\log_{x_1} \\left( x_2 - \\frac{1}{4} \\right) + \\log_{x_2} \\left( x_3 - \\frac{1}{4} \\right) + \\cdots + \\log_{x_{2017}} \\left( x_1 - \\frac{1}{4} \\right) \\right|\n\\]\nwhere \\(x_1, x_2, \\ldots, x_{201... | math | {
"style": "rule",
"ground_truth": "4034"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 31127,
"sample_id": "deepmath-103k:031127",
"difficulty": 8,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions",
"topic_family": "algebra",
"prompt_tokens": 169,
"p0_stage": "truncation-rescued-train",
"gap_label": "tr... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nSquare $ABCD$ has area $200$. Point $E$ lies on side $\\overline{BC}$. Points $F$ and $G$ are the midpoints of $\\overline{AE}$ and $\\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the area of triangle $GCD$?\n\nPlease reason step... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 102177,
"sample_id": "deepmath-103k:102177",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Polygons",
"topic_family": "geometry",
"prompt_tokens": 110,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven the polynomial \\(x^4 + px^3 + qx^2 + rx + s = 0\\) with rational coefficients and four complex roots, two of the roots add up to \\(3 + 4i\\) and the other two have a product of \\(13-i\\). Compute the value of \\(q\\).\n\nPlease reason step by step, and put yo... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 19141,
"sample_id": "deepmath-103k:019141",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 96,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_res... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven 100 lines on a plane, determine the maximum number of right triangles that can be formed by selecting any three of these lines.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62500"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 90883,
"sample_id": "deepmath-103k:090883",
"difficulty": 6.5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 58,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_s... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nThe complex numbers $z$ and $w$ satisfy the system\n\\begin{align*}\nz + \\frac{20i}w &= 5+i, \\\\\nw+\\frac{12i}z &= -4+10i.\n\\end{align*}Find the smallest possible value of $\\vert zw\\vert^2$.\n\nPlease reason step by step, and put your final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 98203,
"sample_id": "deepmath-103k:098203",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers",
"topic_family": "algebra",
"prompt_tokens": 105,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncati... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nConsider a square with vertices labeled as $V = \\{A, B, C, D\\}$. Each vertex can be colored with one of three colors from the set $\\{1, 2, 3\\}$. A coloring is defined as a function $f: V \\rightarrow \\{1, 2, 3\\}$. Two colorings $f$ and $g$ are considered equival... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 72374,
"sample_id": "deepmath-103k:072374",
"difficulty": 5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 201,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFrom the set of integers $\\{1,2,3,\\dots,2009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$.\n... | math | {
"style": "rule",
"ground_truth": "803"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 99934,
"sample_id": "deepmath-103k:099934",
"difficulty": 5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 115,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the smallest positive integer \\( n \\) such that the system of equations \\( \\sin x_1 + \\sin x_2 + \\cdots + \\sin x_n = 0 \\) and \\( \\sin x_1 + 2\\sin x_2 + \\cdots + n\\sin x_n = 100 \\) has a solution.\n\nPlease reason step by step, and put your final ans... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 91757,
"sample_id": "deepmath-103k:091757",
"difficulty": 6,
"topic": "Mathematics -> Precalculus -> Trigonometric Functions",
"topic_family": "algebra",
"prompt_tokens": 106,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_i... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the natural number \\( n \\) such that \\( n^5 = 27^5 + 84^5 + 110^5 + 133^5 \\) using modular arithmetic.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 68649,
"sample_id": "deepmath-103k:068649",
"difficulty": 6.5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 75,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nA right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume.... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 99538,
"sample_id": "deepmath-103k:099538",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes",
"topic_family": "geometry",
"prompt_tokens": 134,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the largest integer \\( n \\) such that \\( 3^n \\) divides every product \\( abc \\), where \\( a, b, c \\) are positive integers satisfying \\( a^2 + b^2 = c^2 \\) and \\( 3 \\mid c \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 25249,
"sample_id": "deepmath-103k:025249",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 88,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind all positive integers \\( n > 1 \\) such that \\( n^2 \\) divides \\( 2^n + 1 \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 42155,
"sample_id": "deepmath-103k:042155",
"difficulty": 6.5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 58,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn three training matches, each lasting 90 minutes, exactly one of the seven players $A_{1}, A_{2}, \\cdots, A_{7}$ is on the field at any moment. The total playing time for each of players $A_{1}, A_{2}, A_{3}, A_{4}$ must be divisible by 7, and for players $A_{5}, A... | math | {
"style": "rule",
"ground_truth": "208806"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 90487,
"sample_id": "deepmath-103k:090487",
"difficulty": 6,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 145,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nThree of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 98452,
"sample_id": "deepmath-103k:098452",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes",
"topic_family": "geometry",
"prompt_tokens": 155,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nEvaluate the expression \\( \\prod _{m=0}^{11}\\prod_{n=0}^{12}\\prod_{p=0}^{12}(x^m+y^n+y^p) \\) modulo 13, where \\( x=e^{\\dfrac{2\\pi i}{12}} \\) and \\( y=e^{\\dfrac{2\\pi i}{13}} \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 58239,
"sample_id": "deepmath-103k:058239",
"difficulty": 8.5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 119,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nSolve the equation for \\(x\\): \\[ 0.2 = 0.819 \\exp(-11.5x) + 0.0975 \\exp(-70.1x) + 0.0325 \\exp(-179x) \\] and find the value of \\(x\\) that is approximately 0.1225.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1225"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 7657,
"sample_id": "deepmath-103k:007657",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 114,
"p0_stage": "truncation-rescued-train",
"gap_label": "tru... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet $u_n$ be the $n^\\text{th}$ term of the sequence\n\\[1,\\,\\,\\,\\,\\,\\,2,\\,\\,\\,\\,\\,\\,5,\\,\\,\\,\\,\\,\\,6,\\,\\,\\,\\,\\,\\,9,\\,\\,\\,\\,\\,\\,12,\\,\\,\\,\\,\\,\\,13,\\,\\,\\,\\,\\,\\,16,\\,\\,\\,\\,\\,\\,19,\\,\\,\\,\\,\\,\\,22,\\,\\,\\,\\,\\,\\,23,\\l... | math | {
"style": "rule",
"ground_truth": "5898"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 100854,
"sample_id": "deepmath-103k:100854",
"difficulty": 5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 403,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_stri... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nThree points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 94890,
"sample_id": "deepmath-103k:094890",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 61,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a $2 \\times 2019$ table, fill each cell with a number such that the top row contains 2019 distinct real numbers, and the bottom row contains the same numbers in a different order. Each column must contain two different numbers whose sum is rational. What is the ma... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93025,
"sample_id": "deepmath-103k:093025",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 105,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the value of \\( \\lim_{N \\to \\infty} \\frac{f(N)}{N} \\), where \\( f(N) \\) is the number of integers \\( 1 \\leq n \\leq N \\) such that the number of divisors of \\( n \\), denoted \\( \\tau(n) \\), is greater than \\( \\ln{N} \\).\n\nPlease reason ste... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 26559,
"sample_id": "deepmath-103k:026559",
"difficulty": 7.5,
"topic": "Mathematics -> Number Theory -> Factorization",
"topic_family": "number_theory",
"prompt_tokens": 113,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_s... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a triangle, let the angle bisectors of angles \\(\\alpha\\) and \\(\\beta\\) intersect the circumcircle at points \\(D\\) and \\(E\\) respectively. Determine the angles formed by the segment \\(DE\\) with the angle bisector of angle \\(\\gamma\\).\n\nPlease reason ... | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 94222,
"sample_id": "deepmath-103k:094222",
"difficulty": 6.5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 86,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_i... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nInside a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. A plane is tangent to these two spheres and intersects the cylinder to form an ellipse. Find the sum of the lengths of the major and... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 86093,
"sample_id": "deepmath-103k:086093",
"difficulty": 6,
"topic": "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes",
"topic_family": "geometry",
"prompt_tokens": 96,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the fourth root of the polynomial $p(x)$ of degree 4 with real coefficients that satisfies the following conditions:\n\n1. $p(x) \\geq 0$ for all $x \\in \\mathbb{R}$.\n2. $p(1) = 0$.\n3. $p(2 + i) = 0$.\n\nIt is known that $(x - 1)$, $(x - (2 + i))$, and $(x - (... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 53017,
"sample_id": "deepmath-103k:053017",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 147,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn right $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 12$, $BC = 35$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\over... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 95419,
"sample_id": "deepmath-103k:095419",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 181,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_re... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of roots of the equation \\( \\overbrace{f(f(\\ldots f}^{10 \\text{ times } f}(x) \\ldots)) + \\frac{1}{2} = 0 \\), where \\( f(x) = |x| - 1 \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 85335,
"sample_id": "deepmath-103k:085335",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 93,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescue... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that\n\\[P(1) = P(3) = P(5) = P(7) = a\\]and\n\\[P(2) = P(4) = P(6) = P(8) = -a.\\]What is the smallest possible value of $a$?\n\nPlease reason step by step, and put your final answer within \\... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 97184,
"sample_id": "deepmath-103k:097184",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 113,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the smallest integer $B$ such that $4B$ is not in the image of Euler's totient function $\\phi(n)$. What is the value of $B$?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 12988,
"sample_id": "deepmath-103k:012988",
"difficulty": 6,
"topic": "Mathematics -> Number Theory -> Factorization",
"topic_family": "number_theory",
"prompt_tokens": 67,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the 2019th positive integer \\( n \\) such that \\( \\binom{2n}{n} \\) is not divisible by 5.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37805"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 89849,
"sample_id": "deepmath-103k:089849",
"difficulty": 7,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 65,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_incons... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nConsider a 20-row matrix with several columns, where each entry is either 0 or 1. The matrix satisfies the following conditions: all columns are unique, and for any two columns, the rows that contain 1s in the same positions do not occur more than twice. Determine the... | math | {
"style": "rule",
"ground_truth": "3820"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 85960,
"sample_id": "deepmath-103k:085960",
"difficulty": 7,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 107,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a circle, quadrilateral $ABCD$ is inscribed such that $AB = BC$ and $AD = 3DC$. Point $R$ is on $BD$ such that $DR = 2RB$. Point $Q$ is on $AR$ such that $\\angle ADQ = \\angle BDQ$. Additionally, $\\angle ABQ + \\angle CBD = \\angle QBD$. Line $AB$ intersects line... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93897,
"sample_id": "deepmath-103k:093897",
"difficulty": 6.5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 145,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the complex number \\(a\\) such that the polynomial \\(x^2-x+a\\) divides \\(x^{13}+x+90\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 11698,
"sample_id": "deepmath-103k:011698",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 61,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind all positive integers \\( n \\) such that \\( \\frac{a(n)}{n} = \\frac{2}{3} \\), where \\( a(n) \\) is the smallest positive integer such that \\( (a(n))! \\) is divisible by \\( n \\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 89545,
"sample_id": "deepmath-103k:089545",
"difficulty": 6.5,
"topic": "Mathematics -> Number Theory -> Factorization",
"topic_family": "number_theory",
"prompt_tokens": 88,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_st... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind all real roots of the equation \\((x+1) \\sqrt{x+2} + (x+6)\\sqrt{x+7} = x^2+7x+12\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 69127,
"sample_id": "deepmath-103k:069127",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 71,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncatio... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of distinct ways to color the 12 edges of a cube using the colors red, blue, and yellow, such that exactly 3 edges are red, 3 edges are blue, and 6 edges are yellow.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 86374,
"sample_id": "deepmath-103k:086374",
"difficulty": 5.5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 78,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_res... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nA line through the point \\(P(l,3)\\) intersects the ellipse \\(\\frac{x^2}{16} + \\frac{y^2}{9} = 1\\) at points \\(A\\) and \\(D\\), and intersects the \\(x\\)-axis and \\(y\\)-axis at points \\(B\\) and \\(C\\) respectively. Given that \\(PA \\cdot PD = PB \\cdot P... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 8266,
"sample_id": "deepmath-103k:008266",
"difficulty": 6,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Angles",
"topic_family": "geometry",
"prompt_tokens": 127,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nWhat is the graph diameter of the 1-skeleton of the omnitruncated $E_8$ polytope?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 2394,
"sample_id": "deepmath-103k:002394",
"difficulty": 9,
"topic": "Mathematics -> Discrete Mathematics -> Graph Theory",
"topic_family": "combinatorics",
"prompt_tokens": 54,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet $a_1 = a_2 = a_3 = 1.$ For $n > 3,$ let $a_n$ be the number of real numbers $x$ such that\n\\[x^4 - 2a_{n - 1} x^2 + a_{n - 2} a_{n - 3} = 0.\\]Compute the sum $a_1 + a_2 + a_3 + \\dots + a_{1000}.$\n\nPlease reason step by step, and put your final answer within ... | math | {
"style": "rule",
"ground_truth": "2329"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 99230,
"sample_id": "deepmath-103k:099230",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 135,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the number of positive integer pairs \\((p, n)\\), where \\(p\\) is a prime number, such that \\[ p^2 = \\frac{(2^n + 1)^5 - 1}{2^n} \\].\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 26259,
"sample_id": "deepmath-103k:026259",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 79,
"p0_stage": "truncation-rescued-train",
"gap_label": "tru... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nHow many natural numbers \\( x \\) satisfy \\( 3 \\cdot 10^{13} < x < 10^{14} \\) such that each digit from 1 to 9 appears exactly once in \\( x \\) and no zeros are adjacent to each other?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "35562240"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 28253,
"sample_id": "deepmath-103k:028253",
"difficulty": 5.5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 89,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_res... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nHow many ways are there to color the squares of a 4x4 chessboard using 4 distinct colors such that no two adjacent squares have the same color, and each row and column contains all 4 colors?\n\nPlease reason step by step, and put your final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 77001,
"sample_id": "deepmath-103k:077001",
"difficulty": 5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 73,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nA function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$ and $S(m, 1) = 1$, along with the recurrence relation $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \\geq 2$ and $n \\geq 2$. Determine the largest integer $k$ such that $2^k$ divides $S(7, 7)$. \\( \\tex... | math | {
"style": "rule",
"ground_truth": "370"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 86529,
"sample_id": "deepmath-103k:086529",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 147,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a sequence of positive integers, each sum of 17 consecutive terms is even, and each sum of 18 consecutive terms is odd. Determine the maximum number of terms such a sequence can have.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 54510,
"sample_id": "deepmath-103k:054510",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Prealgebra -> Integers",
"topic_family": "algebra",
"prompt_tokens": 71,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inconsi... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the limit of the sequence defined by the recurrence relation \\(a_{n}(1-sa_{n-1}^{2})+sa_{n-1}^2-a_{n-1}=0\\), where \\(a_0 > 0\\) and \\(s\\in[0,1]\\). The sequence is known to be decreasing and bounded below by 0.\n\nPlease reason step by step, and put you... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 75426,
"sample_id": "deepmath-103k:075426",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 109,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nHow many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?\n$\\textbf{(A)}\\ 2128 \\qquad\\textbf{(B)}\\ 2148 \\qquad\\textbf{(C)}\\ 2160 \\qquad\\textbf{(D)}\... | math | {
"style": "rule",
"ground_truth": "2148"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 102698,
"sample_id": "deepmath-103k:102698",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Polygons",
"topic_family": "geometry",
"prompt_tokens": 144,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet the real numbers \\(x_{1}, x_{2}, \\cdots, x_{1997}\\) satisfy the following conditions:\n1. \\(-\\frac{1}{\\sqrt{3}} \\leq x_{i} \\leq \\sqrt{3}\\) for \\(i=1,2,\\cdots,1997\\)\n2. \\(x_{1} + x_{2} + \\cdots + x_{1997} = -318 \\sqrt{3}\\)\n\nDetermine the maximum... | math | {
"style": "rule",
"ground_truth": "189548"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 87212,
"sample_id": "deepmath-103k:087212",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 184,
"p0_stage": "truncation-rescued-train",
"gap_label": "tr... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nConsider the sequence \\( a_n \\) defined by the recurrence relations:\n\\[ a_0 = 0, \\quad a_1 = 3, \\]\n\\[ a_n = 8a_{n-1} + 9a_{n-2} + 16 \\text{ for } n \\ge 2. \\]\nDetermine the smallest positive integer \\( h \\) such that \\( a_{n+h} - a_n \\) is divisible by ... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 90764,
"sample_id": "deepmath-103k:090764",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 144,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven three distinct positive integers $a$, $b$, and $c$, all greater than 1, if the product $abc$ divides the expression $(ab - 1)(bc - 1)(ca - 1)$, determine the value of $abc$.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 55472,
"sample_id": "deepmath-103k:055472",
"difficulty": 6,
"topic": "Mathematics -> Number Theory -> Factorization",
"topic_family": "number_theory",
"prompt_tokens": 83,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet \\( A \\) be a set of 100 distinct positive integers. Define the set \\( B = \\left\\{ \\frac{a}{b} \\mid a, b \\in A \\text{ and } a \\neq b \\right\\} \\), and let \\( f(A) \\) denote the number of elements in \\( B \\). Find the sum of the maximum and minimum v... | math | {
"style": "rule",
"ground_truth": "10098"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 94226,
"sample_id": "deepmath-103k:094226",
"difficulty": 7,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 121,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the number of integers $n$ such that $2000 \\leq n \\leq 2010$ and $7$ divides $\\left\\lfloor \\frac{\\sqrt{2}}{4}(1+\\sqrt{2})^n \\right\\rfloor$. Provide your answer as a single integer.\n\nPlease reason step by step, and put your final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 42304,
"sample_id": "deepmath-103k:042304",
"difficulty": 5.5,
"topic": "Mathematics -> Number Theory -> Congruences",
"topic_family": "number_theory",
"prompt_tokens": 99,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued_inco... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nSolve for $x$ in the equation: $$(1+x)^{299} - 124x(1+x)^{299} = 1$$\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 44969,
"sample_id": "deepmath-103k:044969",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 68,
"p0_stage": "truncation-rescued-train",
"gap_label": "tru... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the largest positive integer \\( n \\) such that the intersection \\( A_1 \\cap A_2 \\cap \\cdots \\cap A_n \\neq \\emptyset \\), where \\( A_k = \\{ x \\mid 2^k < x^k + x^{k+1} < 2^{k+1} \\} \\) for \\( k = 1, 2, 3, \\ldots \\).\n\nPlease reason step by step, an... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 5821,
"sample_id": "deepmath-103k:005821",
"difficulty": 6.5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions",
"topic_family": "algebra",
"prompt_tokens": 123,
"p0_stage": "truncation-rescued-train",
"gap_label": "t... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nLet $S$ be a set of 2020 distinct points in the plane. Define \\( M = \\{ P : P \\text{ is the midpoint of } XY \\text{ for some distinct points } X, Y \\text{ in } S \\} \\). Determine the minimum possible number of points in the set \\( M \\).\n\nPlease reason step ... | math | {
"style": "rule",
"ground_truth": "4037"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 85535,
"sample_id": "deepmath-103k:085535",
"difficulty": 6,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 101,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a cube with 7 visible corner cubies, determine the number of different permutations and orientations possible for the hidden edges, given that the visible corners and edges are correctly positioned and oriented.\n\nPlease reason step by step, and put your final ans... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 67497,
"sample_id": "deepmath-103k:067497",
"difficulty": 6,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 67,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the smallest integer \\( n > 4 \\) such that there exists a graph with \\( n \\) vertices, no triangles, and for every pair of non-adjacent vertices, there are exactly two vertices adjacent to both.\n\nPlease reason step by step, and put your final answer wi... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 86474,
"sample_id": "deepmath-103k:086474",
"difficulty": 7,
"topic": "Mathematics -> Discrete Mathematics -> Graph Theory",
"topic_family": "combinatorics",
"prompt_tokens": 77,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescue... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIf \\( x = \\sqrt{2} + \\sqrt{3} + \\sqrt{6} \\) is a root of the polynomial equation \\( x^4 + ax^3 + bx^2 + cx + d = 0 \\), where \\( a, b, c, d \\) are integers, find the value of \\( |a + b + c + d| \\).\n\nPlease reason step by step, and put your final answer wit... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 92447,
"sample_id": "deepmath-103k:092447",
"difficulty": 6,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 110,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nTriangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the in... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 103019,
"sample_id": "deepmath-103k:103019",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"topic_family": "geometry",
"prompt_tokens": 163,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_r... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nA particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's path.\nHow many different paths can the parti... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 95119,
"sample_id": "deepmath-103k:095119",
"difficulty": 5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 114,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the minimum number of zeros of the function \\( f(x) \\) on the interval \\([-21, 21]\\), given that for all real numbers \\( x \\): \\[ f(2-x) = f(2+x) \\] \\[ f(5+x) = f(5-x) \\] and \\( f(0) = 0 \\).\n\nPlease reason step by step, and put your final answe... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93778,
"sample_id": "deepmath-103k:093778",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Intermediate Algebra -> Other",
"topic_family": "algebra",
"prompt_tokens": 109,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of positive integral solutions to the equation \\(x_1 + x_2 + x_3 + x_4 + x_5 = x_1 \\cdot x_2 \\cdot x_3 \\cdot x_4 \\cdot x_5\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 24089,
"sample_id": "deepmath-103k:024089",
"difficulty": 6.5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 86,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncatio... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nA smooth ball with a radius of 1 cm is dipped in blue paint and placed between two concentric spheres with radii of 4 cm and 6 cm. The ball is outside the smaller sphere but inside the larger one. As the ball moves, it leaves a blue-bordered region on the smaller sphe... | math | {
"style": "rule",
"ground_truth": "60.75"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93431,
"sample_id": "deepmath-103k:093431",
"difficulty": 5.5,
"topic": "Mathematics -> Geometry -> Solid Geometry -> 3D Shapes",
"topic_family": "geometry",
"prompt_tokens": 131,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nRegular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\\overline{PA_1},\\overline{PA_2},$ and the minor arc $\\widehat{A_1A_2}$ of the circle has area $\\tfrac{1}{7},$ while the... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 101926,
"sample_id": "deepmath-103k:101926",
"difficulty": 5,
"topic": "Mathematics -> Geometry -> Plane Geometry -> Polygons",
"topic_family": "geometry",
"prompt_tokens": 257,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescued... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nGiven that $b + \\frac{1}{b} + b^2 + \\frac{1}{b^2} + 1 = 0$, where $b$ is a real number, find the value of $b^{4m} + b^{2m} + \\frac{1}{b^{2m}} + \\frac{1}{b^{4m}}$ for any positive integer $m$.\n\nPlease reason step by step, and put your final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 67087,
"sample_id": "deepmath-103k:067087",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 117,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_resc... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of distinct real roots of the polynomial equation \\(x^7 + 20x^6 - x^5 - 576 = 0\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 83815,
"sample_id": "deepmath-103k:083815",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"topic_family": "algebra",
"prompt_tokens": 66,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_rescu... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nIn a game, player A selects a number n from the set S = \\{1, 2, \\ldots, 1001\\}. Player B must guess the value of n by choosing subsets of S. After each guess, player A reveals the number of subsets that contain n. Player B makes three guesses, with the number of su... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 40737,
"sample_id": "deepmath-103k:040737",
"difficulty": 6.5,
"topic": "Mathematics -> Discrete Mathematics -> Combinatorics",
"topic_family": "combinatorics",
"prompt_tokens": 150,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_re... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nFind the largest integer \\( n \\) such that \\((1! + 2! + 3! + \\ldots + n!) + 16\\) is a perfect power.\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 72768,
"sample_id": "deepmath-103k:072768",
"difficulty": 5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 68,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncation_... |
zwhe99/DeepMath-103K/p0-truncation-rescued | [
{
"role": "user",
"content": "Problem:\nDetermine the number of solutions in natural numbers for the equation \\((a+1)(b+1)(c+1)=2abc\\).\n\nPlease reason step by step, and put your final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"source": "zwhe99/DeepMath-103K",
"source_index": 93135,
"sample_id": "deepmath-103k:093135",
"difficulty": 5.5,
"topic": "Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"topic_family": "algebra",
"prompt_tokens": 59,
"p0_stage": "truncation-rescued-train",
"gap_label": "truncatio... |
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