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": "42",
"style": "rule"
} | {
"index": 285,
"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": "2309",
"style": "rule"
} | {
"index": 75,
"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": "84",
"style": "rule"
} | {
"index": 236,
"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": "25636",
"style": "rule"
} | {
"index": 68,
"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": "2n - 8",
"style": "rule"
} | {
"index": 400,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $A=(a_{ij})\\in M_{(n+1)\\times (n+1)}(\\mathbb{R})$ with $a_{ij}=a+|i-j|d$, where $a$ and $d$ are fixed real numbers.\nCalculate $\\det(A)$.",
"role": "user"
}
] | math | {
"ground_truth": "(-1)^n 2^{n-1} d^n (2a + nd)",
"style": "rule"
} | {
"index": 118,
"split": "without-guidance"
} | easy-without-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": 111,
"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": "243",
"style": "rule"
} | {
"index": 271,
"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": "13",
"style": "rule"
} | {
"index": 133,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A circle of radius $3$ is cut into six congruent arcs. These arcs are then rearranged symmetrically to form a hexagonal star as illustrated below. Determine the ratio of the area of the hexagonal star to the area of the original circle.\nA) $\\frac{4.5}{\\pi}$\nB) $\\frac{4.5\\sqrt{2}}{\\pi}$\nC) ... | math | {
"ground_truth": "\\frac{4.5\\sqrt{3}}{\\pi}",
"style": "rule"
} | {
"index": 350,
"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": "7295",
"style": "rule"
} | {
"index": 250,
"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": "71.8",
"style": "rule"
} | {
"index": 69,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement, in which he either claims that one of the three is guilty, or claims that one of the three is innocent. It is possible for two or more of the musketeers to make the same s... | math | {
"ground_truth": "99",
"style": "rule"
} | {
"index": 39,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Some squares of a $n \\times n$ table $(n>2)$ are black, the rest are white. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers.",
"role": "user"
}
] | math | {
"ground_truth": "3n^{2}-5n+2",
"style": "rule"
} | {
"index": 50,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "You start out with a big pile of $3^{2004}$ cards, with the numbers $1,2,3, \\ldots, 3^{2004}$ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these $3^{2003}$ ... | math | {
"ground_truth": "3^{2004}-2 \\cdot 3^{1002}+2",
"style": "rule"
} | {
"index": 223,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:\n- All four rectangles share a common vertex $P$.\n- For each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$.\n- For each $n = 1, 2, 3$, the opposite side of $R_n$ passes through... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 278,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "On a table lies a regular 100-gon whose vertices are numbered consecutively: $1, 2, \\cdots , 100$. These numbers were then rewritten, in increasing order, according to the distance of the corresponding vertex from the front edge of the table. If vertices are at the same distance from the edge, th... | math | {
"ground_truth": "20200",
"style": "rule"
} | {
"index": 134,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "For any positive integer $n, S_{n}$ be the set of all permutations of \\{1,2,3, \\ldots, n\\}. For each permutation $\\pi \\in S_{n}$, let $f(\\pi)$ be the number of ordered pairs $(j, k)$ for which $\\pi(j)>\\pi(k)$ and $1 \\leq j<k \\leq n$. Further define $g(\\pi)$ to be the number of positive ... | math | {
"ground_truth": "995 \\times 2^{998}",
"style": "rule"
} | {
"index": 232,
"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": "0.037",
"style": "rule"
} | {
"index": 166,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 3c$, and $c < 4d$. Additionally, suppose $b + d = 200$. The largest possible value for $a$ is:\nA) 438\nB) 440\nC) 445\nD) 449\nE) 455",
"role": "user"
}
] | math | {
"ground_truth": "449",
"style": "rule"
} | {
"index": 370,
"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": "931",
"style": "rule"
} | {
"index": 144,
"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 \\cdot\\left(3^{2017}-2^{2017}\\right)",
"style": "rule"
} | {
"index": 225,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $m \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.",
"role": "user"
}
] | math | {
"ground_truth": "243",
"style": "rule"
} | {
"index": 126,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000. Otherwise, it will be $25-27 \\frac{|A-C|}{\\min (A, C)-1000}$.",
... | math | {
"ground_truth": "1019",
"style": "rule"
} | {
"index": 79,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $r_{k}$ denote the remainder when $\\binom{127}{k}$ is divided by 8. Compute $r_{1}+2 r_{2}+3 r_{3}+\\cdots+63 r_{63}$.",
"role": "user"
}
] | math | {
"ground_truth": "8096",
"style": "rule"
} | {
"index": 65,
"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": "100,000,000",
"style": "rule"
} | {
"index": 420,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A road company is trying to build a system of highways in a country with $21$ cities. Each highway runs between two cities. A trip is a sequence of distinct cities $C_1,\\dots, C_n$, for which there is a highway between $C_i$ and $C_{i+1}$. The company wants to fulfill the following two constraint... | math | {
"ground_truth": "192",
"style": "rule"
} | {
"index": 410,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Consider a $10 \\times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \\%$ of the burrito's original size and accidental... | math | {
"ground_truth": "71.8",
"style": "rule"
} | {
"index": 69,
"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": 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": "120",
"style": "rule"
} | {
"index": 308,
"split": "with-guidance"
} | easy-with-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": 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": "0.875",
"style": "rule"
} | {
"index": 153,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Triangle $ABC$ has $\\angle C = 60^{\\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\\tan{\\angle BAD}$?\n$\\mathrm{(A)}\\ \\frac{\\sqrt{3}}{6}\\qquad\\mathrm{(B)}\\ \\frac{\\sqrt{3}}{3}\\qquad\\mathrm{(C)}\\ \\frac{\\sqrt{3}}{2\\sqrt{2}}\\qquad\\ma... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{4\\sqrt{2}-3}",
"style": "rule"
} | {
"index": 164,
"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": "315",
"style": "rule"
} | {
"index": 365,
"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{2}{3}",
"style": "rule"
} | {
"index": 0,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "What is the maximum number of rational points that can lie on a circle in $\\mathbb{R}^2$ whose center is not a rational point? (A \\emph{rational point} is a point both of whose coordinates are rational numbers.)",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 102,
"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": "573",
"style": "rule"
} | {
"index": 81,
"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{10}{\\sqrt{3}}",
"style": "rule"
} | {
"index": 339,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Arnold and Kevin are playing a game in which Kevin picks an integer \\(1 \\leq m \\leq 1001\\), and Arnold is trying to guess it. On each turn, Arnold first pays Kevin 1 dollar in order to guess a number \\(k\\) of Arnold's choice. If \\(m \\geq k\\), the game ends and he pays Kevin an additional ... | math | {
"ground_truth": "859",
"style": "rule"
} | {
"index": 103,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "How many squares are shown in the drawing?",
"role": "user"
}
] | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 418,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "20 players are playing in a Super Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays 18 distinct other players exactly once, one is chosen uniformly at random. Find the expected number o... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 74,
"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": "042",
"style": "rule"
} | {
"index": 62,
"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": "1991",
"style": "rule"
} | {
"index": 128,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? Two vertices are nei... | math | {
"ground_truth": "118",
"style": "rule"
} | {
"index": 19,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Two positive integers $p,q \\in \\mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinit... | math | {
"ground_truth": "\\frac{p+q}{\\gcd(p,q)}",
"style": "rule"
} | {
"index": 383,
"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": "149",
"style": "rule"
} | {
"index": 177,
"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": "4^n + 2 \\cdot 3^n - 4 \\cdot 2^n + 1",
"style": "rule"
} | {
"index": 210,
"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": "34",
"style": "rule"
} | {
"index": 241,
"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": "2016",
"style": "rule"
} | {
"index": 156,
"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": "10",
"style": "rule"
} | {
"index": 32,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\\pi/81$. Find the remainder when $N$ is divided by $23$.",
"role": "user"
}
] | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 316,
"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": "4950",
"style": "rule"
} | {
"index": 216,
"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": "506",
"style": "rule"
} | {
"index": 265,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "In each square of a garden shaped like a $2022 \\times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:\n[list]\n[*] The gardener chooses a square in the garden. Each tree on that ... | math | {
"ground_truth": "2271380",
"style": "rule"
} | {
"index": 401,
"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": "319",
"style": "rule"
} | {
"index": 295,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\\frac{... | math | {
"ground_truth": "25517",
"style": "rule"
} | {
"index": 45,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $P_1P_2\\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$.",
"role": "user"
}
] | math | {
"ground_truth": "258",
"style": "rule"
} | {
"index": 6,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \\in \\mathbb{Z}$, she ascends to heaven, but if she lan... | math | {
"ground_truth": "\\frac{13}{22}",
"style": "rule"
} | {
"index": 224,
"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": "100",
"style": "rule"
} | {
"index": 234,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A \\emph{transversal} of an $n\\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \\times n$ matrices $A$ satisfying the following two conditions: \\begin{enumerate} \\item[(a)] Each entry $\\alpha_{i,j}$ of $A$ is in the set $\\{-... | math | {
"ground_truth": "4^n + 2 \\cdot 3^n - 4 \\cdot 2^n + 1",
"style": "rule"
} | {
"index": 210,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $n = 10$. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1 \\leqslant i \\leqslant n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 135,
"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": "216",
"style": "rule"
} | {
"index": 303,
"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": "84",
"style": "rule"
} | {
"index": 297,
"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": "51",
"style": "rule"
} | {
"index": 414,
"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": "13",
"style": "rule"
} | {
"index": 165,
"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": "335",
"style": "rule"
} | {
"index": 52,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. What time does Leo think it is?\n\nA) 11:30 \nB) 11:45 \nC) 12:00 \nD) 1... | math | {
"ground_truth": "12:30",
"style": "rule"
} | {
"index": 173,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possibl... | math | {
"ground_truth": "129",
"style": "rule"
} | {
"index": 233,
"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": "24",
"style": "rule"
} | {
"index": 93,
"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": 380,
"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": "123+6\\sqrt{119}",
"style": "rule"
} | {
"index": 362,
"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": "9",
"style": "rule"
} | {
"index": 321,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Over all pairs of complex numbers $(x, y)$ satisfying the equations $$x+2y^{2}=x^{4} \\quad \\text{and} \\quad y+2x^{2}=y^{4}$$ compute the minimum possible real part of $x$.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt[3]{\\frac{1-\\sqrt{33}}{2}}",
"style": "rule"
} | {
"index": 23,
"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": "3/28",
"style": "rule"
} | {
"index": 151,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Dorothea has a $3 \\times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color.",
"role": "user"
}
] | math | {
"ground_truth": "284688",
"style": "rule"
} | {
"index": 60,
"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": "1512",
"style": "rule"
} | {
"index": 194,
"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": "7",
"style": "rule"
} | {
"index": 219,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $ n$ and $ k$ be positive integers such that $ \\frac{1}{2} n < k \\leq \\frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \\times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.",
... | math | {
"ground_truth": "$4(n-k)$",
"style": "rule"
} | {
"index": 403,
"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": "1991",
"style": "rule"
} | {
"index": 375,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\\overline{A C}$ and $\\overline{A E}$ trisect $\\angle B A F$. (b) $\\overline{B E} \\| \\overline{C D}$ and $\\overline{C F} \\| \\overline{D E}$. (c) $A B=2 A C=4 A E=8 A F$. Suppose that quadrilaterals $A C D E$ and $A D... | math | {
"ground_truth": "7295",
"style": "rule"
} | {
"index": 250,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Compute the number of positive integers less than 10! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials.",
"role": "user"
}
] | math | {
"ground_truth": "648",
"style": "rule"
} | {
"index": 91,
"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": "0.94",
"style": "rule"
} | {
"index": 352,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Li Yun is sitting by the window in a train moving at a speed of 60 km/h. He sees a freight train with 30 cars approaching from the opposite direction. When the head of the freight train passes the window, he starts timing, and he stops timing when the last car passes the window. The recorded time ... | math | {
"ground_truth": "44",
"style": "rule"
} | {
"index": 336,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "In $\\triangle ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Circle $\\omega$ intersects $\\overline{AB}$ at $E$ and $B$, $\\overline{BC}$ at $B$ and $D$, and $\\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\\frac{DG}{EG} = \\frac{3}{4}$, length $DE=\\frac{a\\sqrt{b}}{c}$, where $a$ and $c$ are... | math | {
"ground_truth": "41",
"style": "rule"
} | {
"index": 366,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \\mid a_{2023}$ ?",
"role": "user"
}
] | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 321,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Rectangle $ABCD$ has an area of $32$, and side $\\overline{AB}$ is parallel to the x-axis. Side $AB$ measures $8$ units. Vertices $A,$ $B$, and $C$ are located on the graphs of $y = \\log_a x$, $y = 2\\log_a x$, and $y = 4\\log_a x$, respectively. Determine the value of $a$.\nA) $\\sqrt[3]{\\frac{... | math | {
"ground_truth": "\\sqrt[4]{\\frac{1 + \\sqrt{33}}{2} + 8}",
"style": "rule"
} | {
"index": 176,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \\pi$ and $25 \\pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \\pi$. If $V$ is the volume of $E$, compute $V / \\pi$.... | math | {
"ground_truth": "75",
"style": "rule"
} | {
"index": 28,
"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 $|x_{2012}|?$",
"role": "user"
}
] | math | {
"ground_truth": "84",
"style": "rule"
} | {
"index": 297,
"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{13}",
"style": "rule"
} | {
"index": 184,
"split": "with-guidance"
} | easy-with-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": 202,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Find the smallest integer $k \\geq 2$ such that for every partition of the set $\\{2, 3,\\hdots, k\\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$.",
"role": "user"
}
] | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 108,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P_{}^{}$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertic... | math | {
"ground_truth": "597",
"style": "rule"
} | {
"index": 186,
"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": "6384",
"style": "rule"
} | {
"index": 319,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a > b > c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equa... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 130,
"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": "99956",
"style": "rule"
} | {
"index": 207,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is",
"role": "user"
}
] | math | {
"ground_truth": "38",
"style": "rule"
} | {
"index": 421,
"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": "31",
"style": "rule"
} | {
"index": 422,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A right rectangular prism has edge lengths $\\log_{5}x, \\log_{8}x,$ and $\\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.\nA) $1,000,000$\nB) $10,000,000$\nC) $100,000,000$\nD) $1,000,000,000$\nE) $10,000,000,000$",
"role": "user"
}... | math | {
"ground_truth": "100,000,000",
"style": "rule"
} | {
"index": 420,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "In the diagram, a road network between the homes of five friends is shown. The shortest distance by road from Asya to Galia is 12 km, from Galia to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galia is 15 km, and from Vasya to Galia is 17 km. What is the shortest distance by road from... | math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 162,
"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": "1515",
"style": "rule"
} | {
"index": 411,
"split": "with-guidance"
} | easy-with-guidance |
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