data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | level stringclasses 3
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": "16",
"style": "rule"
} | {
"index": 157,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "An integer-valued function $f$ is called tenuous if $f(x) + f(y) > y^2$ for all positive integers $x$ and $y.$ Let $g$ be a tenuous function such that $g(1) + g(2) + \\dots + g(20)$ is as small as possible. Compute the minimum possible value for $g(14).$",
"role": "user"
}
] | math | {
"ground_truth": "136",
"style": "rule"
} | {
"index": 143,
"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": "23",
"style": "rule"
} | {
"index": 79,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct.",
"role": "user"
}
] | math | {
"ground_truth": "132",
"style": "rule"
} | {
"index": 0,
"split": "without-guidance"
} | easy-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": 131,
"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": "103",
"style": "rule"
} | {
"index": 70,
"split": "with-guidance"
} | medium-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": 149,
"split": "with-guidance"
} | medium-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": 110,
"split": "with-guidance"
} | medium-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": 146,
"split": "with-guidance"
} | easy-with-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"
} | easy-without-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": 61,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Suppose that $3\\triangle_4=\\triangle2_{11}$, where $\\triangle$ represents a digit 0-9. What is the value of the missing digit?",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 127,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy $lcm (a, b, c, d) < 1000$ and $a+b = c+d$, compute the largest possible value of $a + b$.",
"role": "user"
}
] | math | {
"ground_truth": "581",
"style": "rule"
} | {
"index": 96,
"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": "8128",
"style": "rule"
} | {
"index": 105,
"split": "with-guidance"
} | medium-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": "6",
"style": "rule"
} | {
"index": 171,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, ... | math | {
"ground_truth": "14034",
"style": "rule"
} | {
"index": 45,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?",
... | math | {
"ground_truth": "161",
"style": "rule"
} | {
"index": 101,
"split": "without-guidance"
} | easy-without-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"
} | easy-without-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"
} | 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": "581",
"style": "rule"
} | {
"index": 96,
"split": "with-guidance"
} | medium-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": "391",
"style": "rule"
} | {
"index": 164,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\\overline{abcd},$ where at least one of the digits $a,$ $b,$ $c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as fractions in lowest ... | math | {
"ground_truth": "392",
"style": "rule"
} | {
"index": 80,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "In the representation of numbers A and B as decimal fractions, the lengths of their minimal periods are equal to 6 and 12, respectively. What might the length of the minimal period in the similar representation of $A + B$ be? Find all answers and provide the sum of these possible lengths.",
"r... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 158,
"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": "6006",
"style": "rule"
} | {
"index": 104,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "A regular pentagon with area \\(\\sqrt{5}+1\\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form of \\(n - \\sqrt{m}\\). Please find th... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 166,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Imagine a regular a $2015$-gon with edge length $2$. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices... | math | {
"ground_truth": "2012",
"style": "rule"
} | {
"index": 107,
"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": 61,
"split": "with-guidance"
} | easy-with-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"
} | easy-without-guidance |
without-guidance | [
{
"content": "A polynomial $p(x) = \\sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \\ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \\le j \\le n-1,$ and $a_{j+1}-a_j=-k$ for $n \\le j \\le 2n-2;$ we call $... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 118,
"split": "without-guidance"
} | 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": "16902",
"style": "rule"
} | {
"index": 95,
"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": "142",
"style": "rule"
} | {
"index": 120,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\\sqrt{2k}).$ Evaluate $$\\sum_{k=1}^{\\infty}(-1)^{k-1}\\frac{A(k)}{k}.$$ The original answer is in the format $\\frac{m\\pi^j}{n}$, please provide the value of m + n + j.",
"role": "user"
}
] | math | {
"ground_truth": "19",
"style": "rule"
} | {
"index": 154,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?",
"role": "user"
}
] | math | {
"ground_truth": "61",
"style": "rule"
} | {
"index": 58,
"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": "539",
"style": "rule"
} | {
"index": 10,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Daniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each number in Daniel's list, Martin writes the number of times that number appears in Daniel's list, creating a new list of the same length. When Martin's list is read from bottom to t... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 26,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$. What is the maximum possible value of $a + b + c + d + e$?",
"role": "user"
}
] | math | {
"ground_truth": "153",
"style": "rule"
} | {
"index": 51,
"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": "5",
"style": "rule"
} | {
"index": 162,
"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": "319",
"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": "23097",
"style": "rule"
} | {
"index": 29,
"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": "167",
"style": "rule"
} | {
"index": 8,
"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": 40,
"split": "with-guidance"
} | medium-with-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": 119,
"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": "8",
"style": "rule"
} | {
"index": 134,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "What is the smallest positive integer that consists of each of the ten digits in base 10, each used exactly once, and is divisible by each of the digits $2$ through $9$?",
"role": "user"
}
] | math | {
"ground_truth": "1234759680",
"style": "rule"
} | {
"index": 19,
"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": "50",
"style": "rule"
} | {
"index": 132,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in ... | math | {
"ground_truth": "258",
"style": "rule"
} | {
"index": 77,
"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": "103",
"style": "rule"
} | {
"index": 46,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $\\zeta = \\cos \\frac {2pi}{13} + i \\sin \\frac {2pi}{13}$ . Suppose $a > b > c > d$ are positive integers satisfying\n$$|\\zeta^a + \\zeta^b + \\zeta^c +\\zeta^d| =\\sqrt3.$$\nCompute the smallest possible value of $1000a + 100b + 10c + d$.",
"role": "user"
}
] | math | {
"ground_truth": "7521",
"style": "rule"
} | {
"index": 100,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let \n$$P(x) = 24x^{24} + \\sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).$$\n Let $z_{1},z_{2},\\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let \n$$\\sum_{k = 1}^{r}|b_{k}| = m + n\\... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 64,
"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": "21",
"style": "rule"
} | {
"index": 82,
"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": "489",
"style": "rule"
} | {
"index": 123,
"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": "8128",
"style": "rule"
} | {
"index": 1,
"split": "with-guidance"
} | easy-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"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:\n- $P$ passes through $A$ and $B$,\n- the focus $F$ of $P$ lies on $E$,\n- the orthocenter $H$ of $\\triangle FAB$ lies on the directrix of $P$.\n\nIf the major and minor axes of $E$ have lengths $50$ and $... | math | {
"ground_truth": "2402",
"style": "rule"
} | {
"index": 56,
"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": "136",
"style": "rule"
} | {
"index": 143,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.\n(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)",
"role": "user"
}
] | math | {
"ground_truth": "1344",
"style": "rule"
} | {
"index": 144,
"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": "1940",
"style": "rule"
} | {
"index": 88,
"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": "720",
"style": "rule"
} | {
"index": 68,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$. Whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the \"units\" (rightmost) digit, increments this digit, and ... | math | {
"ground_truth": "1727",
"style": "rule"
} | {
"index": 24,
"split": "without-guidance"
} | easy-without-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"
} | 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": 125,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "Let $n$ be a positive integer. When the leftmost digit of the standard base 10 representation of $n$ is shifted to the rightmost position (the units position), the result is $\\frac{n}{3}$. Find the smallest possible value of the sum of the digits of $n$.",
"role": "user"
}
] | math | {
"ground_truth": "126",
"style": "rule"
} | {
"index": 2,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "A square has sides of length 2. Set $\\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.",
"role": "... | math | {
"ground_truth": "86",
"style": "rule"
} | {
"index": 135,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Three circular arcs of radius 5 units bound the region shown. Arcs $AB$ and $AD$ are quarter-circles, and arc $BCD$ is a semicircle. What is the area, in square units, of the region? [asy]\n/* AMC8 2000 #19 Problem */\ndraw((0,0)..(1,1)..(2,0));\ndraw((0,0)..(.7,-.3)..(1,-1));\ndraw((1,-1)..(1.3, ... | math | {
"ground_truth": "50",
"style": "rule"
} | {
"index": 132,
"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": "510",
"style": "rule"
} | {
"index": 75,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "The triangle $ABC$ satisfies $AB=10$ and has angles $\\angle A=75^{\\circ}$, $\\angle B=60^{\\circ}$, and $\\angle C = 45^{\\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter ... | math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 33,
"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 \\mod n)$th hole, where $k \\mod n$ is the remainder when $k$ is divided by $n$. What is the minimum $n$ such ... | math | {
"ground_truth": "326",
"style": "rule"
} | {
"index": 47,
"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": "120",
"style": "rule"
} | {
"index": 103,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. \"Look, daddy!\" she exclaims. \"That number is evenly divisible by the age of each of us kids!\" \"That's right,\... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 172,
"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": "39",
"style": "rule"
} | {
"index": 108,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "Consider the $5 \\times 5 \\times 5$ equilateral triangular grid as shown:\n\n```asy\nsize(5cm);\nreal n = 5;\nfor (int i = 0; i < n; ++i) {\n draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));\n}\nfor (int i = 0; i < n; ++i) {\n draw((n-i,0)--((n-i)/2,(n-i)*0.866));\n}\nfor (int i = 0; i < n; ++i) {... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 27,
"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": "183",
"style": "rule"
} | {
"index": 140,
"split": "with-guidance"
} | medium-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": "6",
"style": "rule"
} | {
"index": 166,
"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",
"style": "rule"
} | {
"index": 170,
"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": "2520",
"style": "rule"
} | {
"index": 89,
"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": 116,
"split": "with-guidance"
} | medium-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": "378",
"style": "rule"
} | {
"index": 78,
"split": "with-guidance"
} | medium-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": 87,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Consider the sequence given by $a_0 = 3$ and such that for $i \\ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.",
"role": "user"
}
] | math | {
"ground_truth": "35",
"style": "rule"
} | {
"index": 115,
"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": "27",
"style": "rule"
} | {
"index": 138,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.",
"role": "user"
}
] | math | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": 89,
"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": "132",
"style": "rule"
} | {
"index": 0,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Find the sum of all the positive integers that have at most three (not necessarily distinct) prime factors, where the primes come from the set \\( \\{ 2, 3, 5, 7 \\} \\).",
"role": "user"
}
] | math | {
"ground_truth": "1932",
"style": "rule"
} | {
"index": 38,
"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": "26",
"style": "rule"
} | {
"index": 109,
"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": 90,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "On the game board below Kendra will start at the center of the board. For each turn she will spin this spinner with four congruent sectors once, and then she will move one space in the direction indicated on the spinner. The \"Start'' square does not have a numerical value, but Kendra may land on ... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 82,
"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": "8",
"style": "rule"
} | {
"index": 163,
"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": "664",
"style": "rule"
} | {
"index": 30,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "$A$ rectangle has sides of length $a$ and $36$. $A$ hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. Whe... | math | {
"ground_truth": "720",
"style": "rule"
} | {
"index": 68,
"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": "101",
"style": "rule"
} | {
"index": 118,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Define the sequence of positive integers $\\{a_n\\}$ as follows:\n- Let $a_1=1$, $a_2=3$.\n- For each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$).\n\nFor example:\n- $a_2=3_{10}=11_2$, so $a_3=1... | math | {
"ground_truth": "23097",
"style": "rule"
} | {
"index": 29,
"split": "without-guidance"
} | 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": "19",
"style": "rule"
} | {
"index": 154,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\\sum^{\\lfloor \\log_{23} n\\rfloor}_{i=1}\ns_{20} \\left( \\left\\lfloor \\frac{n}{23^i} \\right\\rfloor \\right)= 103 \\... | math | {
"ground_truth": "81",
"style": "rule"
} | {
"index": 98,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "In circle $\\theta_1$ with radius $1$, circles $\\phi_1, \\phi_2, \\dots, \\phi_8$, with equal radii, are drawn such that for $1 \\le i \\le 8$, $\\phi_i$ is tangent to $\\omega_1$, $\\phi_{i-1}$, and $\\phi_{i+1}$, where $\\phi_0 = \\phi_8$ and $\\phi_1 = \\phi_9$. There exists a circle $\\omega_... | math | {
"ground_truth": "31",
"style": "rule"
} | {
"index": 12,
"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": "495",
"style": "rule"
} | {
"index": 145,
"split": "with-guidance"
} | medium-with-guidance |
without-guidance | [
{
"content": "What is the smallest square-free composite number that can divide a number of the form $4242\\ldots 42\\pm 1$?",
"role": "user"
}
] | math | {
"ground_truth": "319",
"style": "rule"
} | {
"index": 84,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3).$Let \n$$S=\\sum_{k=1}^{2020}(-1)^{d(k)}k^3.$$ \nDetermine $S$ modulo 2020.",
"role": "user"
}
] | math | {
"ground_truth": "1990",
"style": "rule"
} | {
"index": 149,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Find the number of four-element subsets of $\\{1,2,3,4,\\cdots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subsets.",
"role": "u... | math | {
"ground_truth": "210",
"style": "rule"
} | {
"index": 73,
"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": "161",
"style": "rule"
} | {
"index": 101,
"split": "with-guidance"
} | medium-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": "61",
"style": "rule"
} | {
"index": 58,
"split": "with-guidance"
} | easy-with-guidance |
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