answer string | prompt string | prompt_token_ids list | response list | dataset string | response_token_ids list | response_token_length int64 | correctness int64 | conversations list | template_prompt string | answer_option string | membership int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|
70 | Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$. | [
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"Okay, so I need to find all integer bases b greater than 9 such that 17 base b is a divisor of 97 base b. Then sum all those bases. Hmm, let me start by understanding what the problem is asking.\n\nFirst, I remember that when a number is written in base b, each digit represents a power of b. So, for example, the n... | aime25 | [
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"from": "human",
"value": "Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find all integer bases b greater than 9 such that 1... | Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.
Please reason step by step, and put your final answer within \boxed{}. | null | 0 |
588 | "On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=(...TRUNCATED) | [151644,872,198,1925,57960,55114,19360,3,3501,400,32,27266,42125,54876,323,400,33,3,10246,429,1973,3(...TRUNCATED) | ["Okay, so I need to find the area of heptagon AFNBCEM given some information about triangle ABC and(...TRUNCATED) | aime25 | [32313,11,773,358,1184,311,1477,279,3082,315,566,417,6113,19885,33342,2104,44,2661,1045,1995,911,214(...TRUNCATED) | 14,013 | 1 | [{"from":"human","value":"On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\over(...TRUNCATED) | "On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=(...TRUNCATED) | null | 1 |
16 | "The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a si(...TRUNCATED) | [151644,872,198,785,220,24,3613,315,264,19507,2083,3937,311,458,9853,12644,1346,9566,1283,862,1809,1(...TRUNCATED) | ["Okay, so there's this problem about a baseball team with 9 members who went to an ice cream parlor(...TRUNCATED) | aime25 | [32313,11,773,358,614,419,3491,1588,911,264,19507,2083,448,220,24,3613,2087,311,458,9853,12644,1346,(...TRUNCATED) | 6,620 | 1 | [{"from":"human","value":"The 9 members of a baseball team went to an ice cream parlor after their g(...TRUNCATED) | "The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a si(...TRUNCATED) | null | 0 |
117 | "Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED) | [151644,872,198,9885,279,1372,315,11457,13530,4930,87,7358,15087,11,1380,2176,400,87,3,323,400,88,3,(...TRUNCATED) | ["Okay, so I need to find the number of ordered pairs (x, y) where both x and y are integers between(...TRUNCATED) | aime25 | [32313,11,773,358,1184,311,1477,279,1372,315,11457,13530,320,87,11,379,8,1380,2176,856,323,379,525,2(...TRUNCATED) | 8,748 | 1 | [{"from":"human","value":"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integ(...TRUNCATED) | "Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED) | null | 0 |
279 | "There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ ex(...TRUNCATED) | [151644,872,198,3862,525,400,23,5824,19,15,18,17,15,3,8063,48342,6785,25780,429,990,1817,315,279,185(...TRUNCATED) | ["Okay, so I need to find how many 8-digit numbers formed by permuting the digits 1 through 8 are di(...TRUNCATED) | aime25 | [32313,11,773,358,1184,311,1477,279,1372,315,220,23,48342,5109,1667,1817,15723,504,220,16,311,220,23(...TRUNCATED) | 10,257 | 1 | [{"from":"human","value":"There are $8!=40320$ eight-digit positive integers that use each of the di(...TRUNCATED) | "There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ ex(...TRUNCATED) | null | 0 |
504 | "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | [151644,872,198,2082,374,436,346,642,489,2027,89,588,702,458,1640,17433,12671,68660,311,1817,315,118(...TRUNCATED) | ["Okay, so I need to find r squared plus s squared for an isosceles trapezoid with an inscribed circ(...TRUNCATED) | aime25 | [32313,11,773,358,614,419,3491,911,458,374,436,346,642,489,2027,89,588,448,458,1640,17433,12671,13,5(...TRUNCATED) | 4,439 | 1 | [{"from":"human","value":"An isosceles trapezoid has an inscribed circle tangent to each of its four(...TRUNCATED) | "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | null | 0 |
821 | "The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs of letters.(...TRUNCATED) | [151644,872,198,785,29235,11931,400,32,8161,11289,27266,42125,27229,37306,43539,37768,58862,43654,54(...TRUNCATED) | ["Okay, so I need to find the probability that the last word listed contains the letter G when we ra(...TRUNCATED) | aime25 | [32313,11,773,358,1184,311,1477,279,18927,429,279,1537,3409,10007,5610,279,6524,479,979,582,26618,67(...TRUNCATED) | 22,106 | 1 | [{"from":"human","value":"The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped i(...TRUNCATED) | "The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs of letters.(...TRUNCATED) | null | 1 |
77 | "Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one (...TRUNCATED) | [151644,872,198,10061,400,74,3,387,1931,5109,1741,429,279,1849,400,91,17,20,10,17,15,72,9141,87803,2(...TRUNCATED) | ["Okay, so I need to solve this problem where we have two equations involving complex numbers, and w(...TRUNCATED) | aime25 | [32313,11,1077,594,1490,13,358,1184,311,11625,419,3491,1380,582,614,264,1849,315,37906,15860,6351,51(...TRUNCATED) | 4,267 | 1 | [{"from":"human","value":"Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|(...TRUNCATED) | "Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one (...TRUNCATED) | null | 1 |
62 | "The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin.(...TRUNCATED) | [151644,872,198,785,1346,370,7924,448,23606,400,88,25387,47822,17,19732,19,3,374,45620,400,21,15,61,(...TRUNCATED) | ["Okay, so I need to find the y-coordinate of the intersection point in the fourth quadrant between (...TRUNCATED) | aime25 | [32313,11,773,358,1184,311,1477,279,379,80697,315,279,19246,1459,304,279,11737,93286,1948,279,4024,1(...TRUNCATED) | 11,650 | 1 | [{"from":"human","value":"The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclo(...TRUNCATED) | "The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin.(...TRUNCATED) | null | 1 |
81 | "The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row con(...TRUNCATED) | [151644,872,198,785,220,17,22,7761,315,264,400,18,4955,1733,24,3,5827,525,10199,304,1667,279,5109,22(...TRUNCATED) | ["Okay, so I need to figure out the number of ways to fill a 3x9 grid with numbers 1 through 9, foll(...TRUNCATED) | aime25 | [32313,11,773,358,614,419,3491,911,25009,279,1372,315,5510,311,5155,264,220,18,87,24,5827,448,5109,2(...TRUNCATED) | 20,244 | 0 | [{"from":"human","value":"The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 throu(...TRUNCATED) | "The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row con(...TRUNCATED) | null | 1 |
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