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problem
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| messages
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sequencelengths 32
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Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$. | 70 | 2025-I-1 | "<｜begin▁of▁sentence｜><｜User｜>Find the sum of all integer bases $b>9$ for which $17_{b}$(...TRUNCATED) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["70","70","70","70","70","70","70","70","70","70","70","70","70","70","70","70","70","70","70","70"(...TRUNCATED) | ["Alright, so I have this problem: I need to find the sum of all integer bases \\( b > 9 \\) for whi(...TRUNCATED) | "PROBLEM:\nFind the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.\nSOL(...TRUNCATED) | [0.83203125,0.828125,0.7890625,0.77734375,0.8203125,0.71875,0.7421875,0.64453125,0.8046875,0.8476562(...TRUNCATED) |
"On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=(...TRUNCATED) | 588 | 2025-I-2 | "<｜begin▁of▁sentence｜><｜User｜>On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order(...TRUNCATED) | [1.0,1.0,0.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0(...TRUNCATED) | ["588","588","168","588","588","588","588","108","588","588","702","588","588","588","588","588","58(...TRUNCATED) | ["Okay, so I need to solve this geometry problem, and honestly, it seems a bit complex at first glan(...TRUNCATED) | "PROBLEM:\nOn $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with (...TRUNCATED) | [0.921875,0.921875,0.91796875,0.90625,0.78125,0.90234375,0.85546875,0.85546875,0.75390625,0.75390625(...TRUNCATED) |
"The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a si(...TRUNCATED) | 16 | 2025-I-3 | "<｜begin▁of▁sentence｜><｜User｜>The 9 members of a baseball team went to an ice cream parl(...TRUNCATED) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["16","16","16","16","16","16","16","32","16","292","16","16","16","16","16","16","16","16","16","16(...TRUNCATED) | ["Alright, so I have this problem where 9 baseball team members each get a single scoop of ice cream(...TRUNCATED) | "PROBLEM:\nThe 9 members of a baseball team went to an ice cream parlor after their game. Each playe(...TRUNCATED) | [0.89453125,0.86328125,0.91015625,0.9296875,0.93359375,0.875,0.88671875,0.875,0.87109375,0.94140625,(...TRUNCATED) |
"Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $1(...TRUNCATED) | 117 | 2025-I-4 | "<｜begin▁of▁sentence｜><｜User｜>Find the number of ordered pairs $(x,y)$, where both $x$ a(...TRUNCATED) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["117","117","117","117","117","117","117","117","117","117","117","117","117","117","117","33","117(...TRUNCATED) | ["Okay, so I have this problem: find the number of ordered pairs (x, y) where both x and y are integ(...TRUNCATED) | "PROBLEM:\nFind the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-1(...TRUNCATED) | [0.77734375,0.8515625,0.89453125,0.8203125,0.89453125,0.81640625,0.87109375,0.87890625,0.765625,0.90(...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) | 279 | 2025-I-5 | "<｜begin▁of▁sentence｜><｜User｜>There are $8!=40320$ eight-digit positive integers that us(...TRUNCATED) | [0.0,0.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,0.0,0.0,1.0,1.0,0.0,1.0,1.0,0.0,0.0,0.0(...TRUNCATED) | ["2025","567","279","279","279","135","279","279","279","279","279","279","279","7191","279","7191",(...TRUNCATED) | ["Okay, so I have this problem here where I need to find the number of eight-digit integers that use(...TRUNCATED) | "PROBLEM:\nThere are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5(...TRUNCATED) | [0.66015625,0.765625,0.77734375,0.7890625,0.80859375,0.90625,0.75390625,0.796875,0.85546875,0.738281(...TRUNCATED) |
"An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the(...TRUNCATED) | 504 | 2025-I-6 | "<｜begin▁of▁sentence｜><｜User｜>An isosceles trapezoid has an inscribed circle tangent to (...TRUNCATED) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["504","504","504","504","504","504","504","504","504","504","504","504","504","504","504","504","50(...TRUNCATED) | ["Okay, so I have this problem about an isosceles trapezoid with an inscribed circle. The radius of (...TRUNCATED) | "PROBLEM:\nAn isosceles trapezoid has an inscribed circle tangent to each of its four sides. The rad(...TRUNCATED) | [0.8828125,0.93359375,0.93359375,0.9140625,0.89453125,0.92578125,0.93359375,0.890625,0.85546875,0.89(...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) | 821 | 2025-I-7 | "<｜begin▁of▁sentence｜><｜User｜>The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are ra(...TRUNCATED) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | ["12","271","12","12","271","271","12","12","1","12","12","239","271","1","12","12","12","12","12","(...TRUNCATED) | ["Okay, so I need to find the probability that when the twelve letters A through L are randomly grou(...TRUNCATED) | "PROBLEM:\nThe twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs o(...TRUNCATED) | [0.8515625,0.89453125,0.87890625,0.8515625,0.91796875,0.82421875,0.8203125,0.87109375,0.6796875,0.83(...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) | 77 | 2025-I-8 | "<｜begin▁of▁sentence｜><｜User｜>Let $k$ be real numbers such that the system $|25+20i-z|=5(...TRUNCATED) | [0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["37","77","77","77","77","77","77","77","77","317","69","77","77","77","77","77","77","77","77","77(...TRUNCATED) | ["Okay, so I have this problem where I need to find all real numbers k such that the system of equat(...TRUNCATED) | "PROBLEM:\nLet $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has ex(...TRUNCATED) | [0.80078125,0.73828125,0.8046875,0.7421875,0.73828125,0.63671875,0.73046875,0.8046875,0.7109375,0.77(...TRUNCATED) |
"The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin.(...TRUNCATED) | 62 | 2025-I-9 | "<｜begin▁of▁sentence｜><｜User｜>The parabola with equation $y=x^{2}-4$ is rotated $60^{\\c(...TRUNCATED) | [1.0,1.0,0.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | ["62","62","64","62",null,null,null,null,null,null,null,null,null,"62",null,null,null,null,null,null(...TRUNCATED) | ["Okay, so I need to find the point of intersection in the fourth quadrant where the original parabo(...TRUNCATED) | "PROBLEM:\nThe parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around t(...TRUNCATED) | [0.84765625,0.828125,0.8203125,0.8046875,0.23046875,0.6640625,0.154296875,0.115234375,0.90625,0.0126(...TRUNCATED) |
"The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row con(...TRUNCATED) | 81 | 2025-I-10 | "<｜begin▁of▁sentence｜><｜User｜>The 27 cells of a $3\\times9$ grid are filled in using the(...TRUNCATED) | [0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | ["44","69","129","17","52","74","210","114","148","69","11","61","69","210","207",null,null,null,nul(...TRUNCATED) | ["Alright, so I need to figure out the number of different ways to fill a 3x9 grid that follows the (...TRUNCATED) | "PROBLEM:\nThe 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that ea(...TRUNCATED) | [0.70703125,0.75,0.390625,0.66015625,0.640625,0.71484375,0.6640625,0.7109375,0.6953125,0.67578125,0.(...TRUNCATED) |
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Generate 32 solutions for aime25 using deepseek-ai/DeepSeek-R1-Distill-Qwen-7B
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