data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | level stringclasses 6
values |
|---|---|---|---|---|---|
without-guidance | [
{
"content": "Let \\(a, b, c\\) be positive integers. All the roots of each of the quadratics \\(a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c\\) are integers. Over all triples \\((a, b, c)\\), find the triple with the third smallest value of \\(a+b+c\\).",
"role": "user"
}
] | math | {
"ground_truth": "(1,10,24)",
"style": "rule"
} | {
"index": 435,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\text{not a rational number}",
"style": "rule"
} | {
"index": 40,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "720",
"style": "rule"
} | {
"index": 68,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "-2 \\sqrt {3}",
"style": "rule"
} | {
"index": 341,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "(n-2)2^n + 1",
"style": "rule"
} | {
"index": 316,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Find the smallest integer $n$ such that each subset of $\\{1,2,\\ldots, 2004\\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$.",
"role": "user"
}
] | math | {
"ground_truth": "1003",
"style": "rule"
} | {
"index": 209,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.",
"role": "user"
}
] | math | {
"ground_truth": "1940",
"style": "rule"
} | {
"index": 88,
"split": "without-guidance"
} | DAPO-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 178,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Given triangle $ABC$, $\\overrightarrow{CA}•\\overrightarrow{CB}=1$, the area of the triangle is $S=\\frac{1}{2}$,<br/>$(1)$ Find the value of angle $C$;<br/>$(2)$ If $\\sin A\\cos A=\\frac{{\\sqrt{3}}}{4}$, $a=2$, find $c$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": 432,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "2^{k-1}",
"style": "rule"
} | {
"index": 377,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \\angle PAB \\equal{} \\angle CQB \\equal{} 45^\\circ$, $ \\angle ABP \\equal{} \\angle QBC \\equal{} 75^\\circ$, $ \\angle RAC \\equal{} 105^\\... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 13,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "A Skolem sequence of order $n$ is a sequence $\\left(s_{1}, s_{2}, \\ldots, s_{2 n}\\right)$ of $2 n$ integers satisfying the conditions:\n\ni) for every $k$ in $\\{1,2,3, \\ldots, n\\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and\n\nii) if $s_{i}=s_{j}=k$ with ... | math | {
"ground_truth": "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)",
"style": "rule"
} | {
"index": 471,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 433,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Compute the number of ways to tile a $3\\times5$ rectangle with one $1\\times1$ tile, one $1\\times2$ tile, one $1\\times3$ tile, one $1\\times4$, and one $1\\times5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)",
"role": "user"
... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 97,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "John is planning to fence a rectangular garden such that the area is at least 150 sq. ft. The length of the garden should be 20 ft longer than its width. Additionally, the total perimeter of the garden must not exceed 70 ft. What should the width, in feet, be?",
"role": "user"
}
] | math | {
"ground_truth": "-10 + 5\\sqrt{10}",
"style": "rule"
} | {
"index": 177,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "142",
"style": "rule"
} | {
"index": 413,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "2 \\left( \\frac{4^n - 1}{3} \\right) + 1",
"style": "rule"
} | {
"index": 23,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "149",
"style": "rule"
} | {
"index": 69,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "For any integer $k \\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p... | math | {
"ground_truth": "149",
"style": "rule"
} | {
"index": 69,
"split": "without-guidance"
} | DAPO-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 167,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "1344",
"style": "rule"
} | {
"index": 200,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "98",
"style": "rule"
} | {
"index": 141,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "2 - \\sqrt{3}",
"style": "rule"
} | {
"index": 418,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Find the number of arrangements of 4 beads (2 red, 2 green, 2 blue) in a circle such that the two red beads are not adjacent.",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 431,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Let\n\\[ x^6 - 3x^3 - x^2 - x - 2 = q_1(x) q_2(x) \\dotsm q_m(x), \\]\nwhere each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(3) + q_2(3) + \\dots + q_m(3)$.",
"role": "user"
}
] | math | {
"ground_truth": "634",
"style": "rule"
} | {
"index": 286,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"index": 69,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "The cross-section of a right cylinder is an ellipse, with semi-axes $a$ and $b$, where $a > b$. The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of ... | math | {
"ground_truth": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": 478,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "A $k \\times k$ array contains each of the numbers $1, 2, \\dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \\in \\mathbb{N}^+$)?",
"role": "user"
}
] | math | {
"ground_truth": "3^{n+1} - 1",
"style": "rule"
} | {
"index": 382,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "1+\\sqrt{\\frac{7}{15}}",
"style": "rule"
} | {
"index": 318,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "On the evening, more than $\\frac{1}{3}$ of the students of a school are going to the cinema. On the same evening, more than $\\frac{3}{10}$ are going to the theatre, and more than $\\frac{4}{11}$ are going to the concert. At least how many students are there in this school?",
"role": "user"
... | math | {
"ground_truth": "173",
"style": "rule"
} | {
"index": 23,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centi... | math | {
"ground_truth": "\\sqrt{\\frac{2160}{\\pi}}",
"style": "rule"
} | {
"index": 419,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\frac{38\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 358,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Find the three-digit positive integer $n$ for which $\\binom n3 \\binom n4 \\binom n5 \\binom n6 $ is a perfect square.",
"role": "user"
}
] | math | {
"ground_truth": "489",
"style": "rule"
} | {
"index": 123,
"split": "without-guidance"
} | DAPO-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 225,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "n - k",
"style": "rule"
} | {
"index": 300,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Two adjacent faces of a tetrahedron, each of which is a regular triangle with a side length of 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 242,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "128",
"style": "rule"
} | {
"index": 137,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": 312,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 156,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality:\n\\[\\left| \\sum_{a \\in A} x^a - \\sum_{b \\in B} x^b \\right| < (1-x)^{2011}.\\]",
"role": "user"
... | math | {
"ground_truth": "\\text{Yes}",
"style": "rule"
} | {
"index": 39,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "129",
"style": "rule"
} | {
"index": 443,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "(x,y,z)=\\left(\\frac{b+c}{2},\\frac{a+c}{2},\\frac{a+b}{2}\\right)",
"style": "rule"
} | {
"index": 223,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "2004",
"style": "rule"
} | {
"index": 42,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\[\n6n - 9\n\\]",
"style": "rule"
} | {
"index": 406,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $S$ be the set of points $(x, y)$ in the Cartesian plane that satisfy\n\\[\\Big|\\big| |x|-2\\big|-1\\Big|+\\Big|\\big| |y|-2\\big|-1\\Big|=1.\\]What is the total length of all the lines that make up $S$?",
"role": "user"
}
] | math | {
"ground_truth": "64\\sqrt{2}",
"style": "rule"
} | {
"index": 291,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?",
"role": "user"
}
] | math | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"index": 378,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "4250",
"style": "rule"
} | {
"index": 4,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Triangle $A B C$ satisfies $\\angle B>\\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\\angle A D M=68^{\\circ}$ ... | math | {
"ground_truth": "86^{\\circ}",
"style": "rule"
} | {
"index": 321,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 414,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "731",
"style": "rule"
} | {
"index": 63,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 242,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.\n(i) The value of $a_0$ is a positive integer.\n(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\\frac{a_i}{a_i + 2} $\n(iii) There exists a positive integer $k... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 135,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.\n- The Local Area Inspirational Math Exam consists of 15 problems.\n- The Further Away Regional Math League has 10 problems.\n- The Distance-Optimized Math Open has 50 problems.\... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 17,
"split": "without-guidance"
} | DAPO-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "(0, 0, 2), (1, 1, 2), (2, 2, 3)",
"style": "rule"
} | {
"index": 269,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "64\\sqrt{2}",
"style": "rule"
} | {
"index": 291,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, ... | math | {
"ground_truth": "14034",
"style": "rule"
} | {
"index": 45,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Find the minimum possible value of $\\sqrt{58-42 x}+\\sqrt{149-140 \\sqrt{1-x^{2}}}$ where $-1 \\leq x \\leq 1$",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{109}",
"style": "rule"
} | {
"index": 265,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "In a regular quadrilateral frustum with lateral edges \\(A A_{1}, B B_{1}, C C_{1}, D D_{1}\\), the side length of the upper base \\(A_{1} B_{1} C_{1} D_{1}\\) is 1, and the side length of the lower base is 7. A plane passing through the edge \\(B_{1} C_{1}\\) perpendicular to the plane \\(A D_{1}... | math | {
"ground_truth": "\\frac{38\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 358,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Consider a sequence of complex numbers $\\left\\{z_{n}\\right\\}$ defined as \"interesting\" if $\\left|z_{1}\\right|=1$ and for every positive integer $n$, the following holds:\n$$\n4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.\n$$\n\nFind the largest constant $C$ such that for any \"interesting\" s... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 416,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 215,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "(3,-24)",
"style": "rule"
} | {
"index": 186,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "86^{\\circ}",
"style": "rule"
} | {
"index": 321,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applicati... | math | {
"ground_truth": "n",
"style": "rule"
} | {
"index": 310,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Determine whether $\\sqrt{1001^2+1}+\\sqrt{1002^2+1}+ \\cdots + \\sqrt{2000^2+1}$ be a rational number or not?",
"role": "user"
}
] | math | {
"ground_truth": "\\text{not a rational number}",
"style": "rule"
} | {
"index": 40,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "1866",
"style": "rule"
} | {
"index": 237,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 67,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 127,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\\equiv n \\pmod m$... | math | {
"ground_truth": "\\text{No}",
"style": "rule"
} | {
"index": 69,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two c... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 402,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 467,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "162",
"style": "rule"
} | {
"index": 20,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containing this edge.",
... | math | {
"ground_truth": "\\frac{9\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 245,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 135,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\sqrt{109}",
"style": "rule"
} | {
"index": 265,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\\angle PAB + \\angle PBC + \\angle PCA... | math | {
"ground_truth": "\\text{the incenter, circumcenter, and orthocenter of } \\triangle ABC",
"style": "rule"
} | {
"index": 25,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "$\\frac{\\sqrt{2}}{3}$",
"style": "rule"
} | {
"index": 464,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Define the sequence of positive integers $\\{a_n\\}$ as follows:\n- Let $a_1=1$, $a_2=3$.\n- For each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$).\n\nFor example:\n- $a_2=3_{10}=11_2$, so $a_3=1... | math | {
"ground_truth": "23097",
"style": "rule"
} | {
"index": 29,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move ... | math | {
"ground_truth": "\\text{Charlie}",
"style": "rule"
} | {
"index": 182,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 171,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its 4 digits when expressed in mmdd format (e.g. the digit sum of 13 May is $0+5+1+3=9$). Find the number of dates in the year 2021 with digit sum equal to the positive integer $n$.",
"role": "user"
}
] | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 225,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\frac{2 \\sqrt{170}-9 \\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 383,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \\geq t_0$ by the following properties: \\begin{enumerate} \\item[(a)] $f(t)$ is continuous for $t \\geq t_0$, and is twice dif... | math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 425,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "391",
"style": "rule"
} | {
"index": 164,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\\geq1.$ Compute\n\n$$\\prod\\limits_{k=0}^{\\infty}\\left(1-\\frac{1}{a_k}\\right)$$\n\nin closed form.The answer is in the form \\frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.",
"role": "user"
}
] | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 153,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "The sides of rhombus \\( EFGH \\) are the hypotenuses of the isosceles right triangles \\( EAF, FDG, GCH, \\) and \\( HBE \\), and all these triangles have common interior points with the rhombus \\( EFGH \\). The sum of the areas of quadrilateral \\( ABCD \\) and rhombus \\( EFGH \\) is 12. Find ... | math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": 228,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal c... | math | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": 165,
"split": "without-guidance"
} | omni-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "114",
"style": "rule"
} | {
"index": 3,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "A polynomial $p(x) = \\sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \\ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \\le j \\le n-1,$ and $a_{j+1}-a_j=-k$ for $n \\le j \\le 2n-2;$ we call $... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 118,
"split": "without-guidance"
} | DAPO-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 287,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "The king summons two wizards into the palace. He demands Wizard A to write down 100 positive real numbers on cards (allowing duplicates) without revealing them to Wizard B. Then, Wizard B must accurately write down all of these 100 positive real numbers. Otherwise, both wizards will be beheaded. W... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 467,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\infty",
"style": "rule"
} | {
"index": 482,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 416,
"split": "with-guidance"
} | ALL-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\text{No solution}",
"style": "rule"
} | {
"index": 315,
"split": "with-guidance"
} | omni-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": 228,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $p(q(x))=q(p(x))$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$.",
"role": "user"
}
] | math | {
"ground_truth": "(3,-24)",
"style": "rule"
} | {
"index": 186,
"split": "without-guidance"
} | ALL-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 118,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 139,
"split": "with-guidance"
} | DAPO-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "\\sqrt{\\frac{2160}{\\pi}}",
"style": "rule"
} | {
"index": 419,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "In convex quadrilateral $ ABCD$, $ AB\\equal{}a$, $ BC\\equal{}b$, $ CD\\equal{}c$, $ DA\\equal{}d$, $ AC\\equal{}e$, $ BD\\equal{}f$. If $ \\max \\{a,b,c,d,e,f \\}\\equal{}1$, then find the maximum value of $ abcd$.",
"role": "user"
}
] | math | {
"ground_truth": "2 - \\sqrt{3}",
"style": "rule"
} | {
"index": 418,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Tim is doing a cakewalk with $11$ steps. He takes his first step on step $1$ and takes a total of $139$ steps, walking in a circle (so that after the 11th step he reaches the first step again). Which step of the cakewalk does he end on?",
"role": "user"
}
] | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 139,
"split": "without-guidance"
} | DAPO-without-guidance |
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