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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) | ["Okay, so I have this problem here: I need to find the sum of all integer bases \\( b > 9 \\) for w(...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) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,0.0,0.0,0.0(...TRUNCATED) | ["588","588","588","588","588","588","588","240","588","588","588","588","588","588","588","588","58(...TRUNCATED) | ["Alright, so we have this geometry problem here about triangle ABC. On side AB, there are points A,(...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) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["16","16","16","16","16","16","16","16","16","16","16","16","16","16","16","16","16","16","16","16"(...TRUNCATED) | ["Alright, so I've got this problem here about a baseball team with 9 members who went to an ice cre(...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) | [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0(...TRUNCATED) | ["117","117","117","117","117","117","117","117","117","117","117","117","117","117","117","117","11(...TRUNCATED) | ["Alright, so I need to find the number of ordered pairs (x, y) where both x and y are integers betw(...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) | [1.0,1.0,0.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,0.0,0.0,1.0,0.0,0.0,1.0,1.0,1.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | ["279","279","7191","279","567","279","279","279","279","279","279","2025","423","423","279","9","13(...TRUNCATED) | ["Okay, so I have this problem where I need to find the number of eight-digit integers that use each(...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) |
"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","311","12","1","41","18","107","100","1","1","1","949","701","319","949","1","1","1/11","577",(...TRUNCATED) | ["Okay, so I have this problem here with twelve letters: A, B, C, D, E, F, G, H, I, J, K, and L. The(...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) | [1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,0.0(...TRUNCATED) | ["77","77","77","77","77","69","77","77","77","77","77","77","77","77","77","77","77","77","31","77"(...TRUNCATED) | ["Okay, so I have this problem where I need to find the sum of all possible values of \\( k \\) such(...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) | [1.0,1.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0(...TRUNCATED) | ["62","62","44","62","62","62","62","62","62","62",null,null,null,null,null,null,null,null,null,null(...TRUNCATED) | ["Alright, so I have this problem where I need to find the unique intersection point in the fourth q(...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) | ["45","136","39","102","17","69","91","210","1440","17","17","38","17","69","44","135","120",null,nu(...TRUNCATED) | ["Okay, so I need to figure out the number of different ways to fill a 3x9 grid with numbers 1 throu(...TRUNCATED) |
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