data_source stringclasses 2
values | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict | level stringclasses 2
values |
|---|---|---|---|---|---|
without-guidance | [
{
"content": "Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer. \nHow many triples $(a, b, c)$ satisfy the conditions?",
"role": "user"
}
] | math | {
"ground_truth": "16",
"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": "98",
"style": "rule"
} | {
"index": 287,
"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": 136,
"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 |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "715",
"style": "rule"
} | {
"index": 262,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Triangle $ABC$ is inscribed in circle $\\omega$. Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\\frac{m}{n}$, where $m$ and $n$ are relatively prime... | math | {
"ground_truth": "43",
"style": "rule"
} | {
"index": 122,
"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": "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 $23$.\n",
"role": "user"
}
] | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 208,
"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": 295,
"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": 78,
"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": "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": "1",
"style": "rule"
} | {
"index": 252,
"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": "581",
"style": "rule"
} | {
"index": 178,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\\pi$. Find $n$.",
"role": "user"
}
] | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 13,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point \\(B\\) from the line on which the... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 342,
"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": "161",
"style": "rule"
} | {
"index": 186,
"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": 16,
"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": "5",
"style": "rule"
} | {
"index": 326,
"split": "with-guidance"
} | easy-with-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": 178,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $a$ and $b$ be positive integers such that $1 \\leq a < b \\leq 100$. If there exists a positive integer $k$ such that $ab \\mid a^k + b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs.",
"role": "user"
}
] | math | {
"ground_truth": "96",
"style": "rule"
} | {
"index": 71,
"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": "392",
"style": "rule"
} | {
"index": 136,
"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": "23",
"style": "rule"
} | {
"index": 314,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A cube of side 3 inches has a cube of side 1 inch cut from each corner. A cube of side 2 inches is then inserted in each corner. What is the number of square inches in the surface area of the resulting solid?",
"role": "user"
}
] | math | {
"ground_truth": "198",
"style": "rule"
} | {
"index": 274,
"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": "6",
"style": "rule"
} | {
"index": 333,
"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": "2012",
"style": "rule"
} | {
"index": 197,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $... | math | {
"ground_truth": "16902",
"style": "rule"
} | {
"index": 175,
"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": "734",
"style": "rule"
} | {
"index": 83,
"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": 227,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\\frac{n}{6}$ are drawn such that $A$, $C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $\\triangle AJG$ is an eq... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 59,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?",
"role": "user"
... | math | {
"ground_truth": "142",
"style": "rule"
} | {
"index": 234,
"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": "11",
"style": "rule"
} | {
"index": 304,
"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": "61",
"style": "rule"
} | {
"index": 96,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "For any integer $k \\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p... | math | {
"ground_truth": "149",
"style": "rule"
} | {
"index": 119,
"split": "without-guidance"
} | easy-without-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 48,
"split": "with-guidance"
} | easy-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "153",
"style": "rule"
} | {
"index": 84,
"split": "with-guidance"
} | easy-with-guidance |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 327,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "$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": 117,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "For an integer $30 \\le k \\le 70$, let $M$ be the maximum possible value of:\n\\[ \\frac{A}{\\gcd(A,B)} \\]\nwhere $A = \\binom{100}{k}$ and $B = \\binom{100}{k+3}$. Find the remainder when $M$ is divided by $1000$.",
"role": "user"
}
] | math | {
"ground_truth": "664",
"style": "rule"
} | {
"index": 56,
"split": "without-guidance"
} | easy-without-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": 292,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made duri... | math | {
"ground_truth": "1167",
"style": "rule"
} | {
"index": 163,
"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": "279",
"style": "rule"
} | {
"index": 230,
"split": "with-guidance"
} | easy-with-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": 272,
"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": "1234759680",
"style": "rule"
} | {
"index": 36,
"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 |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then complete the solution from where it left off. Ensure your continuation is logically consistent and leads to a complete and correc... | math | {
"ground_truth": "258",
"style": "rule"
} | {
"index": 132,
"split": "with-guidance"
} | easy-with-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": 79,
"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": "15",
"style": "rule"
} | {
"index": 104,
"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": 141,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.",
"role": "user"
}
] | math | {
"ground_truth": "490",
"style": "rule"
} | {
"index": 65,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Compute the smallest integer $n\\geq 4$ such that $\\textstyle\\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\\textstyle\\binom n4$ are $0000$).",
"role": "user"
}
] | math | {
"ground_truth": "8128",
"style": "rule"
} | {
"index": 192,
"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": "114",
"style": "rule"
} | {
"index": 11,
"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": "2402",
"style": "rule"
} | {
"index": 94,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Rhombus \\(ABCD\\) has side length \\(2\\) and \\(\\angle B = 120^\\circ\\). Region \\(R\\) consists of all points inside the rhombus that are closer to vertex \\(B\\) than any of the other three vertices. The area of \\(R\\) is given in the form \\(\\frac{k\\sqrt{n}}{m}\\). Please find the value ... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 327,
"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": "10",
"style": "rule"
} | {
"index": 288,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.",
"role": "user"
}
] | math | {
"ground_truth": "1940",
"style": "rule"
} | {
"index": 158,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$. Then the area of $S$ has th... | math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 113,
"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": "1727",
"style": "rule"
} | {
"index": 43,
"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": "1344",
"style": "rule"
} | {
"index": 292,
"split": "with-guidance"
} | easy-with-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": 252,
"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": "731",
"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": "50",
"style": "rule"
} | {
"index": 263,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Let $\\triangle ABC$ be an acute triangle with circumcircle $\\omega,$ and let $H$ be the intersection of the altitudes of $\\triangle ABC.$ Suppose the tangent to the circumcircle of $\\triangle HBC$ at $H$ intersects $\\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\\tr... | math | {
"ground_truth": "58",
"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": "29",
"style": "rule"
} | {
"index": 60,
"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": "326",
"style": "rule"
} | {
"index": 79,
"split": "with-guidance"
} | easy-with-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 |
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 |
without-guidance | [
{
"content": "Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$) and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk... | math | {
"ground_truth": "770",
"style": "rule"
} | {
"index": 85,
"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": 49,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the heigh... | math | {
"ground_truth": "183",
"style": "rule"
} | {
"index": 286,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $a_1 = 1$ and for $k > 1$ define $a_k=[\\sqrt{a_1+a_2+\\cdots+a_{k-1}} ],$where $[x]$ denotes the integer part of x. Find $a_{1000}$.",
"role": "user"
}
] | math | {
"ground_truth": "495",
"style": "rule"
} | {
"index": 293,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Two \\(4 \\times 4\\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \\(k - m... | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 341,
"split": "without-guidance"
} | easy-without-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": 161,
"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": 84,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that \n$\\frac{x^2}{a^2} + \\frac{y^2}{a^2-16} = \\frac{(x-20)^2}{b^2-1} + \\frac{(y-11)^2}{b^2} = 1.$ \n Find the least possible value of $a+b.$",
"role": "user"
}
] | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 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": "30",
"style": "rule"
} | {
"index": 341,
"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": "35",
"style": "rule"
} | {
"index": 221,
"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 |
with-guidance | [
{
"content": "You are given a problem and a partial solution.\n\nYour task is to carefully study the partial response, identify what reasoning or steps are already provided, and then 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": 161,
"split": "with-guidance"
} | easy-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": 333,
"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": "1932",
"style": "rule"
} | {
"index": 67,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Call a positive integer $n$ [i]tubular [/i] if for any two distinct primes $p$ and $q$ dividing $n, (p + q) | n$. Find the number of tubular numbers less than $100,000$. (Integer powers of primes, including $1, 3$, and $16$, are not considered [i]tubular[/i].)",
"role": "user"
}
] | math | {
"ground_truth": "26",
"style": "rule"
} | {
"index": 204,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "\t Define the sequence $a_1, a_2 \\dots$ as follows: $a_1=1$ and for every $n\\ge 2$,\n\\[\na_n = \\begin{cases}\nn-2 & \\text{if } a_{n-1} =0 \\\\\na_{n-1} -1 & \\text{if } a_{n-1} \\neq 0\n\\end{cases}\n\\]\nA non-negative integer $d$ is said to be {\\em jet-lagged} if there are non-negative int... | math | {
"ground_truth": "51",
"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": "21",
"style": "rule"
} | {
"index": 141,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "On the evening, more than $\\frac{1}{3}$ of the students of a school are going to the cinema. On the same evening, more than $\\frac{3}{10}$ are going to the theatre, and more than $\\frac{4}{11}$ are going to the concert. At least how many students are there in this school?",
"role": "user"
... | math | {
"ground_truth": "173",
"style": "rule"
} | {
"index": 42,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that $\\frac{||1|| + ||2|| + ...+ ||n||}{n}=100$.",
"ro... | math | {
"ground_truth": "89800",
"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": "720",
"style": "rule"
} | {
"index": 117,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "A polynomial $p(x) = \\sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \\ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \\le j \\le n-1,$ and $a_{j+1}-a_j=-k$ for $n \\le j \\le 2n-2;$ we call $... | math | {
"ground_truth": "101",
"style": "rule"
} | {
"index": 227,
"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": "23097",
"style": "rule"
} | {
"index": 52,
"split": "with-guidance"
} | easy-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": 308,
"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": "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": "31",
"style": "rule"
} | {
"index": 26,
"split": "with-guidance"
} | easy-with-guidance |
without-guidance | [
{
"content": "Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be se... | math | {
"ground_truth": "520",
"style": "rule"
} | {
"index": 127,
"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": 145,
"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": 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": "210",
"style": "rule"
} | {
"index": 126,
"split": "with-guidance"
} | easy-with-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": 186,
"split": "without-guidance"
} | easy-without-guidance |
without-guidance | [
{
"content": "How many real numbers $x$ satisfy the equation $\\frac{1}{5}\\log_2 x = \\sin (5\\pi x)$ ?",
"role": "user"
}
] | math | {
"ground_truth": "159",
"style": "rule"
} | {
"index": 98,
"split": "without-guidance"
} | easy-without-guidance |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 22