data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | level stringclasses 2
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": "335",
"style": "rule"
} | {
"index": 1245,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "In the diagram, points $U$, $V$, $W$, $X$, $Y$, and $Z$ lie on a straight line with $UV=VW=WX=XY=YZ=5$. Semicircles with diameters $UZ$, $UV$, $VW$, $WX$, $XY$, and $YZ$ create the shape shown. What is the area of the shaded region?\n[asy]\nsize(5cm); defaultpen(fontsize(9));\npair one = (1, 0);... | math | {
"ground_truth": "\\frac{325}{4}\\pi",
"style": "rule"
} | {
"index": 1024,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Compute the remainder when $99989796\\ldots 121110090807 \\ldots 01$ is divided by $010203 \\ldots 091011 \\ldots 9798$. Note that the first sequence starts at $99$, and the second sequence ends at $98$.",
"role": "user"
}
] | math | {
"ground_truth": "9801",
"style": "rule"
} | {
"index": 82,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2 + qx + r =0$. Which primes appear in seven or more elements of $S$? Compute the product of all the valid primes as the final answer.",
"role": "user"
}
] | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 1617,
"split": "without-guidance"
} | easy-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": "342",
"style": "rule"
} | {
"index": 100,
"split": "with-guidance"
} | easy-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": "81",
"style": "rule"
} | {
"index": 180,
"split": "with-guidance"
} | easy-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": 90,
"split": "with-guidance"
} | easy-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": "181",
"style": "rule"
} | {
"index": 1619,
"split": "with-guidance"
} | easy-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": "96",
"style": "rule"
} | {
"index": 302,
"split": "with-guidance"
} | easy-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": "30(1 + \\sqrt{5})",
"style": "rule"
} | {
"index": 1118,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A student did not notice a multiplication sign between two three-digit numbers and wrote one 6-digit number that happened to be 7 times greater than the product of the two three-digit numbers. Find the factors. ",
"role": "user"
}
] | math | {
"ground_truth": "143",
"style": "rule"
} | {
"index": 298,
"split": "without-guidance"
} | easy-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": "1430",
"style": "rule"
} | {
"index": 637,
"split": "with-guidance"
} | easy-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": "101",
"style": "rule"
} | {
"index": 1489,
"split": "with-guidance"
} | easy-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": 223,
"split": "with-guidance"
} | easy-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": 1009,
"split": "with-guidance"
} | easy-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": "153",
"style": "rule"
} | {
"index": 84,
"split": "with-guidance"
} | easy-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 \\sqrt{3} / 2",
"style": "rule"
} | {
"index": 948,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "For a prime $q$, let $\\Phi_q(x) = x^{q-1} + x^{q-2} + \\cdots + x + 1$. \nFind the sum of all primes $p$ such that $3 \\le p \\le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying\n\\[\\binom{N}{\\Phi_q(p)} \\equiv \\binom{2\\Phi_q(p)}{N} \\not\\equiv 0 \\pmod{p}.\\]",
... | math | {
"ground_truth": "420",
"style": "rule"
} | {
"index": 28,
"split": "without-guidance"
} | easy-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": "33 \\sqrt{3}",
"style": "rule"
} | {
"index": 922,
"split": "with-guidance"
} | easy-with-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": 342,
"split": "without-guidance"
} | easy-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": 1108,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?\n",
"role": "user"
}
] | math | {
"ground_truth": "120",
"style": "rule"
} | {
"index": 190,
"split": "without-guidance"
} | easy-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": "40/21",
"style": "rule"
} | {
"index": 1153,
"split": "with-guidance"
} | easy-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": "457",
"style": "rule"
} | {
"index": 15,
"split": "with-guidance"
} | easy-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": 252,
"split": "with-guidance"
} | easy-with-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": 1606,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \\geqslant 2$ there exists $1 \\leqslant k \\leqslant n$ satisfying\n\n$$\na_{n}=\\frac{a_{n-1}+\\cdots+a_{n-k}}{k}\n$$\n\nFind the maximal possible value of $a_{2018}-a_{2017}$.",
"... | math | {
"ground_truth": "$\\frac{2016}{2017^{2}}$",
"style": "rule"
} | {
"index": 1542,
"split": "without-guidance"
} | easy-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": "126",
"style": "rule"
} | {
"index": 9,
"split": "with-guidance"
} | easy-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{1}{512}",
"style": "rule"
} | {
"index": 533,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "What is the least number of weights in the set that can be divided into either $4, 5$, or 6 piles of equal mass?",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 297,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. ... | math | {
"ground_truth": "279",
"style": "rule"
} | {
"index": 230,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \\geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\\left|x_{2012}\\right|$ ?",
"role": "user"
}
] | math | {
"ground_truth": "84",
"style": "rule"
} | {
"index": 755,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 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 a plane containing the given edge... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 659,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\\frac{N}{M}$.",
"role": "user"
}
... | math | {
"ground_truth": "\\frac{3 \\sqrt{3}+3}{2}",
"style": "rule"
} | {
"index": 1266,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "The radius of the Earth measures approximately $6378 \\mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.\n\nNext, suppose that the length of the wire is increased by $1 \\mathrm{~m}$, so that the wire and the Equator are concentric circles around the Earth. Can... | math | {
"ground_truth": "0.159",
"style": "rule"
} | {
"index": 1042,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $n \\geq 3$ be an odd number and suppose that each square in a $n \\times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squ... | math | {
"ground_truth": "\\left(\\frac{n+1}{2}\\right)^2 + 1",
"style": "rule"
} | {
"index": 955,
"split": "without-guidance"
} | easy-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\\sqrt{3}",
"style": "rule"
} | {
"index": 584,
"split": "with-guidance"
} | easy-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": "1990",
"style": "rule"
} | {
"index": 300,
"split": "with-guidance"
} | easy-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": 807,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in ... | math | {
"ground_truth": "\\frac{2^{n}-1}{8^{n}}",
"style": "rule"
} | {
"index": 1061,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Given the circle $x^2 + y^2 = 4$ and the point $\\\\mathrm{P}(2,1)$, two mutually perpendicular lines are drawn through point $\\\\mathrm{P}$, intersecting the circle at points $\\\\mathrm{A}, \\\\mathrm{B}$ and $\\\\mathrm{C}, \\\\mathrm{D}$ respectively. Point $\\\\mathrm{A}$ lies inside the lin... | math | {
"ground_truth": "$\\sqrt{15}$",
"style": "rule"
} | {
"index": 1486,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Suppose there are $160$ pigeons and $n$ holes. The $1$ st pigeon flies to the $1$ st hole, the $2$ nd pigeon flies to the $4$ th hole, and so on, such that the $i$ th pigeon flies to the $(i^2\\text{ mod }n)$ th hole, where $k\\text{ mod }n$ is the remainder when $k$ is divided by $... | math | {
"ground_truth": "326",
"style": "rule"
} | {
"index": 699,
"split": "without-guidance"
} | easy-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": "14034",
"style": "rule"
} | {
"index": 77,
"split": "with-guidance"
} | easy-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": "32",
"style": "rule"
} | {
"index": 255,
"split": "with-guidance"
} | easy-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": "31",
"style": "rule"
} | {
"index": 781,
"split": "with-guidance"
} | easy-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{9\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": 695,
"split": "with-guidance"
} | easy-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": 46,
"split": "with-guidance"
} | easy-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{3}",
"style": "rule"
} | {
"index": 1420,
"split": "with-guidance"
} | easy-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{41}{144}",
"style": "rule"
} | {
"index": 591,
"split": "with-guidance"
} | easy-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": 415,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Calculate:\n\n1. $(1)(\\sqrt{2}+2)^{2}$\n2. $(2)(\\sqrt{3}-\\sqrt{8})-\\frac{1}{2}(\\sqrt{18}+\\sqrt{12})$",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{7\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": 615,
"split": "without-guidance"
} | easy-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": 328,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the firs... | math | {
"ground_truth": "342",
"style": "rule"
} | {
"index": 283,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $ABC$ be a triangle with sides $AB = 34$, $BC = 15$, $AC = 35$ and let $\\Gamma$ be the circle of smallest possible radius passing through $A$ tangent to $BC$. Let the second intersections of $\\Gamma$ and sides $AB$, $AC$ be the points $X, Y$ . Let the ray $XY$ intersect the circumcircle of t... | math | {
"ground_truth": "173",
"style": "rule"
} | {
"index": 213,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is always the same, regardl... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 807,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?",... | math | {
"ground_truth": "\\frac{253}{256}",
"style": "rule"
} | {
"index": 1197,
"split": "without-guidance"
} | easy-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": "7521",
"style": "rule"
} | {
"index": 184,
"split": "with-guidance"
} | easy-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": "21",
"style": "rule"
} | {
"index": 141,
"split": "with-guidance"
} | easy-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>\\frac{1}{2} \\text{ or } x<-\\frac{1}{3}",
"style": "rule"
} | {
"index": 720,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Given that $\\{x_1, x_2, \\ldots, x_n\\} = \\{1, 2, \\ldots, n\\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \\geq 2$), of \\[ x_1x_2 + x_2x_3 + \\cdots + x_{n-1}x_n + x_nx_1. \\]",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2n^3 + 3n^2 - 11n + 18}{6}",
"style": "rule"
} | {
"index": 1609,
"split": "without-guidance"
} | easy-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}}{4}",
"style": "rule"
} | {
"index": 1407,
"split": "with-guidance"
} | easy-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": 287,
"split": "with-guidance"
} | easy-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": "5n+1",
"style": "rule"
} | {
"index": 1182,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "In how many ways can the set of ordered pairs of integers be colored red and blue such that for all $a$ and $b$, the points $(a, b),(-1-b, a+1)$, and $(1-b, a-1)$ are all the same color?",
"role": "user"
}
] | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 448,
"split": "without-guidance"
} | easy-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": 1080,
"split": "with-guidance"
} | easy-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": "17",
"style": "rule"
} | {
"index": 356,
"split": "with-guidance"
} | easy-with-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": 1188,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a function such that for any integers $x, y$, we have $f\\left(x^{2}-3 y^{2}\\right)+f\\left(x^{2}+y^{2}\\right)=2(x+y) f(x-y)$. Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \\cdot f(2016)$ is a perfect square. Find the minimum possible val... | math | {
"ground_truth": "246",
"style": "rule"
} | {
"index": 479,
"split": "without-guidance"
} | easy-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": "28",
"style": "rule"
} | {
"index": 1310,
"split": "with-guidance"
} | easy-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{\\dfrac{425}{2}}",
"style": "rule"
} | {
"index": 1414,
"split": "with-guidance"
} | easy-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": "\\varnothing",
"style": "rule"
} | {
"index": 590,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$",
"role": "user"
}
] | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 268,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "In regular pentagon $ABCDE$, let $O \\in CE$ be the center of circle $\\Gamma$ tangent to $DA$ and $DE$. $\\Gamma$ meets $DE$ at $X$ and $DA$ at $Y$ . Let the altitude from $B$ meet $CD$ at $P$. If $CP = 1$, the area of $\\vartriangle COY$ can be written in the form $\\frac{a}{b} \\frac{\\sin c^o... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 201,
"split": "without-guidance"
} | easy-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "$m n-\\lfloor m / 2\\rfloor$",
"style": "rule"
} | {
"index": 1529,
"split": "with-guidance"
} | easy-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": "\\begin{pmatrix} 4/7 \\\\ 5/7 \\\\ 8/7 \\end{pmatrix}",
"style": "rule"
} | {
"index": 947,
"split": "with-guidance"
} | easy-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": "7521",
"style": "rule"
} | {
"index": 675,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle.",
"role": "user"
}
] | math | {
"ground_truth": "221 \\sqrt{3} - 330",
"style": "rule"
} | {
"index": 785,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "For each positive integer $n$, there is a circle around the origin with radius $n$. Rainbow Dash starts off somewhere on the plane, but not on a circle. She takes off in some direction in a straight path. She moves \\frac{\\sqrt{5}}{5}$ units before crossing a circle, then \\sqrt{5}$ units, then \... | math | {
"ground_truth": "\\frac{2 \\sqrt{170}-9 \\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 1156,
"split": "without-guidance"
} | easy-without-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": 314,
"split": "without-guidance"
} | easy-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": "148",
"style": "rule"
} | {
"index": 247,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Find the sum of all the $x$, $y$ values in the pairs of real numbers $(x,y)$ satisfying the system of equations \\begin{align*} \\frac{1}{x} + \\frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\\\ \\frac{1}{x} - \\frac{1}{2y} &= 2(y^4-x^4). \\end{align*}",
"role": "user"
}
] | math | {
"ground_truth": "3^{1/5}",
"style": "rule"
} | {
"index": 1613,
"split": "without-guidance"
} | easy-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": 444,
"split": "without-guidance"
} | easy-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": "1121",
"style": "rule"
} | {
"index": 1462,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of 20... | math | {
"ground_truth": "7174",
"style": "rule"
} | {
"index": 1469,
"split": "without-guidance"
} | easy-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": "71",
"style": "rule"
} | {
"index": 937,
"split": "with-guidance"
} | easy-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": "65/3",
"style": "rule"
} | {
"index": 676,
"split": "with-guidance"
} | easy-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{3 \\sqrt{3}+3}{2}",
"style": "rule"
} | {
"index": 1266,
"split": "with-guidance"
} | easy-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": 64,
"split": "without-guidance"
} | easy-without-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 $x_{n+1}X(x_n)=x_np(x_n)$ for $... | math | {
"ground_truth": "149",
"style": "rule"
} | {
"index": 1019,
"split": "without-guidance"
} | easy-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": "\\binom{2n}{k} - 2\\binom{n}{k}",
"style": "rule"
} | {
"index": 980,
"split": "with-guidance"
} | easy-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{38\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 1065,
"split": "with-guidance"
} | easy-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": "2018",
"style": "rule"
} | {
"index": 433,
"split": "with-guidance"
} | easy-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": "490",
"style": "rule"
} | {
"index": 65,
"split": "with-guidance"
} | easy-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",
"style": "rule"
} | {
"index": 272,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. Find the additional length of track required to reduce the grade from $3\\%$ to $2\\%$. If the answer is $x$ ft., what is the value ... | math | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": 337,
"split": "without-guidance"
} | easy-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": "\\sqrt{\\frac{2160}{\\pi}}",
"style": "rule"
} | {
"index": 1288,
"split": "with-guidance"
} | easy-with-guidance |
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