data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | level stringclasses 6
values |
|---|---|---|---|---|---|
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": "32",
"style": "rule"
} | {
"index": 198,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$. Then the area of $S$ has th... | math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 67,
"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": "715",
"style": "rule"
} | {
"index": 131,
"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": "2001",
"style": "rule"
} | {
"index": 343,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Is there a finite abelian group $G$ such that the product of the\norders of all its elements is $2^{2009}$?",
"role": "user"
}
] | math | {
"ground_truth": "No",
"style": "rule"
} | {
"index": 104,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\\circ$ angle, and the straight sides form a $45^\\circ$ angle with the ground. The artwork has a... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 167,
"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": "\\left(\\frac{n+1}{2}\\right)^2 + 1",
"style": "rule"
} | {
"index": 328,
"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": "(1,10,24)",
"style": "rule"
} | {
"index": 435,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)--(-45,0)--(-60,26)--cycle);\ndraw((0,0)--(30,0)--(15,26)... | math | {
"ground_truth": "54",
"style": "rule"
} | {
"index": 126,
"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": "10",
"style": "rule"
} | {
"index": 402,
"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": "14400",
"style": "rule"
} | {
"index": 192,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could h... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 178,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Given a right triangle \\( \\triangle ABC \\) with legs \\( AC=12 \\), \\( BC=5 \\). Point \\( D \\) is a moving point on the hypotenuse \\( AB \\). The triangle is folded along \\( CD \\) to form a right dihedral angle \\( A-CD-B \\). When the length of \\( AB \\) is minimized, let the plane angl... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 198,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Triangle $ABC$ 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 interior of $\\triangle ABC$ are... | math | {
"ground_truth": "715",
"style": "rule"
} | {
"index": 316,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.",
"role": "user"
}
] | math | {
"ground_truth": "(2, 1), (3, 1), (1, 2), (1, 3)",
"style": "rule"
} | {
"index": 19,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak po... | math | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": 192,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$?",
"role": "user"
}
] | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 433,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "From the set of integers $\\{1,2,3,\\dots,3009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$.",
"role": "user"
}
] | math | {
"ground_truth": "1203",
"style": "rule"
} | {
"index": 372,
"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": "51",
"style": "rule"
} | {
"index": 90,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were ... | math | {
"ground_truth": "(m, n) = (36, 6)",
"style": "rule"
} | {
"index": 386,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "In the figure, $\\triangle ABC$ has $\\angle A =45^{\\circ}$ and $\\angle B =30^{\\circ}$. A line $DE$, with $D$ on $AB$ \nand $\\angle ADE =60^{\\circ}$, divides $\\triangle ABC$ into two pieces of equal area.\n(Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$\nThe r... | math | {
"ground_truth": "\\frac{1}{\\sqrt[4]{12}}",
"style": "rule"
} | {
"index": 306,
"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": "199",
"style": "rule"
} | {
"index": 202,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Compute the unique positive integer $m$ such that\n\\[1 \\cdot 2^1 + 2 \\cdot 2^2 + 3 \\cdot 2^3 + \\dots + m \\cdot 2^m = 2^{m + 8}.\\]",
"role": "user"
}
] | math | {
"ground_truth": "129",
"style": "rule"
} | {
"index": 443,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \\cdot CD = AB \\cdot DA$. What is the largest possible value of $BD$?",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{\\dfrac{425}{2}}",
"style": "rule"
} | {
"index": 440,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Let $n \\ge 2$ be an integer, and let $A_n$ be the set \\[A_n = \\{2^n - 2^k\\mid k \\in \\mathbb{Z},\\, 0 \\le k < n\\}.\\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .\n\n[i]",
"role": "user"
}
] | math | {
"ground_truth": "(n-2)2^n + 1",
"style": "rule"
} | {
"index": 316,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Compute the sum of all positive integers $n < 200$ such that $\\text{gcd}(n, k) \\neq 1$ for every $k \\in \\{2 \\cdot 11 \\cdot 19, 3 \\cdot 13 \\cdot 17, 5 \\cdot 11 \\cdot 13, 7 \\cdot 17 \\cdot 19\\}$.",
"role": "user"
}
] | math | {
"ground_truth": "539",
"style": "rule"
} | {
"index": 10,
"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": "\\frac{1}{\\sqrt{2}}",
"style": "rule"
} | {
"index": 478,
"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": "10",
"style": "rule"
} | {
"index": 153,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Given a sequence \\( a_1, a_2, a_3, \\ldots, a_n \\) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \\( n \\)?",
"role": "user"
}
] | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 215,
"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": "n",
"style": "rule"
} | {
"index": 310,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.",
"role": "user"
}
] | math | {
"ground_truth": "132",
"style": "rule"
} | {
"index": 72,
"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": "\\frac{1}{\\sqrt[4]{12}}",
"style": "rule"
} | {
"index": 306,
"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": "715",
"style": "rule"
} | {
"index": 316,
"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": "504",
"style": "rule"
} | {
"index": 296,
"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{2}}{2}",
"style": "rule"
} | {
"index": 392,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Some portions of the line $y=4 x$ lie below the curve $y=10 \\pi \\sin ^{2} x$, and other portions lie above the curve. Compute the sum of the lengths of all the segments of the graph of $y=4 x$ that lie in the first quadrant, below the graph of $y=10 \\pi \\sin ^{2} x$.",
"role": "user"
}
] | math | {
"ground_truth": "$\\frac{5 \\pi}{4} \\sqrt{17}$",
"style": "rule"
} | {
"index": 477,
"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": "2018",
"style": "rule"
} | {
"index": 165,
"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": "4",
"style": "rule"
} | {
"index": 167,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The ... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 156,
"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": "622",
"style": "rule"
} | {
"index": 304,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $a_0$, $a_1$, $a_2$, $\\dots$ be an infinite sequence of real numbers such that $a_0 = \\frac{4}{5}$ and \n\\[ \na_{n} = 2 a_{n-1}^2 - 1 \n\\] \nfor every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satis... | math | {
"ground_truth": "167",
"style": "rule"
} | {
"index": 8,
"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": "173",
"style": "rule"
} | {
"index": 23,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?"... | math | {
"ground_truth": "94",
"style": "rule"
} | {
"index": 35,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Find the total number of different integers the function\n\n\\[ f(x) = \\left\\lfloor x \\right\\rfloor + \\left\\lfloor 2 \\cdot x \\right\\rfloor + \\left\\lfloor \\frac{5 \\cdot x}{3} \\right\\rfloor + \\left\\lfloor 3 \\cdot x \\right\\rfloor + \\left\\lfloor 4 \\cdot x \\right\\rfloor \\]\n\n... | math | {
"ground_truth": "734",
"style": "rule"
} | {
"index": 50,
"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": "No",
"style": "rule"
} | {
"index": 104,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that \n$\\frac{x^2}{a^2} + \\frac{y^2}{a^2-16} = \\frac{(x-20)^2}{b^2-1} + \\frac{(y-11)^2}{b^2} = 1.$ \n Find the least possible value of $a+b.$",
"role": "user"
}
] | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 79,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:\n\n- $M$ is the sum of the squares of some $n$ consecutive positive integers.\n- $2M$ is the sum of the squares of some $2n$ consecutive positive integers.",
"role": "user"
}
] | math | {
"ground_truth": "4250",
"style": "rule"
} | {
"index": 4,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_{1}$ attacks another American $A_{2}$, then $A_{2}$ also attacks $A_{1}$. Let $m$ be the number of squares that an American attacks when placed in the top ... | math | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": 360,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determi... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 263,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Define the sequences $(a_n),(b_n)$ by\n\\begin{align*}\n& a_n, b_n > 0, \\forall n\\in\\mathbb{N_+} \\\\ \n& a_{n+1} = a_n - \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{a_i}} \\\\ \n& b_{n+1} = b_n + \\frac{1}{1+\\sum_{i=1}^n\\frac{1}{b_i}}\n\\end{align*}\n1) If $a_{100}b_{100} = a_{101}b_{101}$, find the ... | math | {
"ground_truth": "199",
"style": "rule"
} | {
"index": 202,
"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": "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)",
"style": "rule"
} | {
"index": 471,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n\\geq 2$. Find the least odd prime factor of $a_{2015}$.",
"role": "user"
}
] | math | {
"ground_truth": "181",
"style": "rule"
} | {
"index": 488,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "There are $256$ players in a tennis tournament ranked from $1$ to $256$, where $1$ is the highest rank and $256$ is the lowest rank. In a match, the player with the higher rank wins with a probability of $\\frac{3}{5}$. \n\nIn each round, players are paired such that the highest-ranked player face... | math | {
"ground_truth": "103",
"style": "rule"
} | {
"index": 46,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "In tetrahedron $ABCD$, triangle $ABC$ is an equilateral triangle, $\\\\angle BCD=90^{\\\\circ}$, $BC=CD=1$,$AC=\\\\sqrt{3}$,$E$ and $F$ are the midpoints of edges $BD$ and $AC$ respectively. Then the cosine of the angle formed by lines $AE$ and $BF$ is \\\\_\\\\_\\\\_\\\\_\\\\_.",
"role": "use... | math | {
"ground_truth": "$\\frac{\\sqrt{2}}{3}$",
"style": "rule"
} | {
"index": 464,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "A sequence of convex polygons $\\{P_n\\}, n \\geq 0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the \\textit{interior} trisection points of the sides of $P_n$. T... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{7}",
"style": "rule"
} | {
"index": 481,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Alice, Bob, and Charlie are playing a game with 6 cards numbered 1 through 6. Each player is dealt 2 cards uniformly at random. On each player's turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob deci... | math | {
"ground_truth": "\\frac{2}{15}",
"style": "rule"
} | {
"index": 227,
"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": "19",
"style": "rule"
} | {
"index": 154,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "In the figure, $m\\angle A = 28^{\\circ}$, $m\\angle B = 74^\\circ$ and $m\\angle C = 26^{\\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x + y$? [asy]\nsize(150);\ndraw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));\ndraw((0,5)--(2,2)--(5,5)--(1... | math | {
"ground_truth": "128",
"style": "rule"
} | {
"index": 137,
"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": "\\frac{\\sqrt{3}}{12}",
"style": "rule"
} | {
"index": 449,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Let $\\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\\frac{\\sqrt{a}-b}{c}$ where $a$ , ... | math | {
"ground_truth": "622",
"style": "rule"
} | {
"index": 304,
"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": "x=\\frac{1}{2}\\pm \\frac {\\sqrt 3} {2} i, n = 2+6k\\ (k\\ge 0)",
"style": "rule"
} | {
"index": 356,
"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": "1024",
"style": "rule"
} | {
"index": 360,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "For an integer $30 \\le k \\le 70$, let $M$ be the maximum possible value of:\n\\[ \\frac{A}{\\gcd(A,B)} \\]\nwhere $A = \\binom{100}{k}$ and $B = \\binom{100}{k+3}$. Find the remainder when $M$ is divided by $1000$.",
"role": "user"
}
] | math | {
"ground_truth": "664",
"style": "rule"
} | {
"index": 30,
"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": "\\text{all even positive integers}",
"style": "rule"
} | {
"index": 323,
"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": "\\frac{2}{3}",
"style": "rule"
} | {
"index": 13,
"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": "1920",
"style": "rule"
} | {
"index": 330,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is th... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 146,
"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": "1940",
"style": "rule"
} | {
"index": 88,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted.",
"role": "user"
}
] | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 421,
"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": "-10 + 5\\sqrt{10}",
"style": "rule"
} | {
"index": 177,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "A small block with an initial speed $v_0$ smoothly moves onto a sloped big block of mass $M$. After the small block reaches a height $h$ on the slope, it slides back down. Find the height $h$ in terms of $v_0$, $g$, $m$, and $M$. The original answer is in the form \\frac{k}{m} \\cdot \\frac{n v_0^... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 162,
"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": "\\text{No}",
"style": "rule"
} | {
"index": 378,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Find the least positive integer $n$ for which there exists a set $\\{s_1, s_2, \\ldots , s_n\\}$ consisting of $n$ distinct positive integers such that\n\\[ \\left( 1 - \\frac{1}{s_1} \\right) \\left( 1 - \\frac{1}{s_2} \\right) \\cdots \\left( 1 - \\frac{1}{s_n} \\right) = \\frac{51}{2010}.\\]\n\... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 287,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is $200$ and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle.",
"role": "... | math | {
"ground_truth": "56",
"style": "rule"
} | {
"index": 151,
"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": "23097",
"style": "rule"
} | {
"index": 29,
"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": "n=3^{1991}",
"style": "rule"
} | {
"index": 345,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\\angle F_{1}PF_{2}=60^{\\circ}$, determine the eccentricity of the ellipse in this pair of related curves.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": 261,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Find all positive integers $n$ such that the set $\\{n,n+1,n+2,n+3,n+4,n+5\\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.",
"role": "user"
}
] | math | {
"ground_truth": "\\text{No solution}",
"style": "rule"
} | {
"index": 315,
"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": "2n - 3 \\text{ if } n \\text{ is odd, and } 2n - 4 \\text{ if } n \\text{ is even}",
"style": "rule"
} | {
"index": 202,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "How many ways are there to color the vertices of a cube using the colors red, blue, or green such that no edge connects two vertices of the same color? Note that rotations and reflections are considered distinct colorings.",
"role": "user"
}
] | math | {
"ground_truth": "114",
"style": "rule"
} | {
"index": 3,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Let $n$ be a positive integer, $n \\geq 2$, and consider the polynomial equation\n\\[x^n - x^{n-2} - x + 2 = 0.\\]\nFor each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$",
"role": "user"
}
] | math | {
"ground_truth": "x=\\frac{1}{2}\\pm \\frac {\\sqrt 3} {2} i, n = 2+6k\\ (k\\ge 0)",
"style": "rule"
} | {
"index": 356,
"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": "664",
"style": "rule"
} | {
"index": 30,
"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{2}",
"style": "rule"
} | {
"index": 263,
"split": "with-guidance"
} | omni-with-guidance |
without-guidance | [
{
"content": "Given \\( x_{0} > 0 \\), \\( x_{0} \\neq \\sqrt{3} \\), a point \\( Q\\left( x_{0}, 0 \\right) \\), and a point \\( P(0, 4) \\), the line \\( PQ \\) intersects the hyperbola \\( x^{2} - \\frac{y^{2}}{3} = 1 \\) at points \\( A \\) and \\( B \\). If \\( \\overrightarrow{PQ} = t \\overrightarrow{QA}... | math | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": 392,
"split": "without-guidance"
} | ALL-without-guidance |
without-guidance | [
{
"content": "Find all of the positive real numbers like $ x,y,z,$ such that :\n\n1.) $ x \\plus{} y \\plus{} z \\equal{} a \\plus{} b \\plus{} c$\n\n2.) $ 4xyz \\equal{} a^2x \\plus{} b^2y \\plus{} c^2z \\plus{} abc$\n\nProposed to Gazeta Matematica in the 80s by VASILE C?RTOAJE and then by Titu Andreescu to I... | 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": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Define the sequence $a_1, a_2 \\dots$ as follows: $a_1=1$ and for every $n\\ge 2$,\n\\[\na_n = \\begin{cases}\nn-2 & \\text{if } a_{n-1} =0 \\\\\na_{n-1} -1 & \\text{if } a_{n-1} \\neq 0\n\\end{cases}\n\\]\nA non-negative integer $d$ is said to be {\\em jet-lagged} if there are non-negative intege... | math | {
"ground_truth": "51",
"style": "rule"
} | {
"index": 90,
"split": "without-guidance"
} | DAPO-without-guidance |
without-guidance | [
{
"content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \\(B\\) from the line on which the... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 171,
"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": "60",
"style": "rule"
} | {
"index": 290,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Trodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peas... | math | {
"ground_truth": "1920",
"style": "rule"
} | {
"index": 330,
"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": "6",
"style": "rule"
} | {
"index": 7,
"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": "14",
"style": "rule"
} | {
"index": 283,
"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": "489",
"style": "rule"
} | {
"index": 123,
"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": "12",
"style": "rule"
} | {
"index": 194,
"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": "11",
"style": "rule"
} | {
"index": 431,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.",
"role": "user"
}
] | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 167,
"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": "634",
"style": "rule"
} | {
"index": 286,
"split": "with-guidance"
} | ALL-with-guidance |
without-guidance | [
{
"content": "Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:\n\n[list]\n\n [*]If at some stage the cards are in $m$ piles, she chooses $1\\le... | math | {
"ground_truth": "2^{n-2}",
"style": "rule"
} | {
"index": 191,
"split": "without-guidance"
} | omni-without-guidance |
without-guidance | [
{
"content": "Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\\triangle ABC$. Lines $\\ell_1$, $\\ell_2$, and $\\ell_3$ pass through $O$ and are parallel to $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$, respectively. The intersecti... | math | {
"ground_truth": "280",
"style": "rule"
} | {
"index": 48,
"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": "35",
"style": "rule"
} | {
"index": 115,
"split": "with-guidance"
} | DAPO-with-guidance |
without-guidance | [
{
"content": "Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\\angle ASB = 30^\\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$.",
"role": "user"
}... | math | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": 293,
"split": "without-guidance"
} | ALL-without-guidance |
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