question string | generations string | labels int64 | full_token_ids list | reasoning_token_ids list | blockwise_attentions list |
|---|---|---|---|---|---|
G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of $c$. | First, I'll analyze the number 1234xy, which is a six-digit number where x and y are unknown digits.
To determine if the number is divisible by 8, I need to look at the last three digits, which are 4xy. For a number to be divisible by 8, the number formed by its last three digits must be divisible by 8. So, 400 + 10x ... | 1 | [
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8. Find $\sqrt{\frac{x}{63}-32} \times \sqrt{\frac{y}{63}-32}$, given that $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$.
ANSWER: 32. | Okay, so I need to find the value of √(x/63 - 32) multiplied by √(y/63 - 32), given that 1/x + 1/y = 1/2016. Hmm, let me see. I remember that when dealing with square roots multiplied together, it might be helpful to combine them under a single square root or perhaps square both sides of the equation to eliminate the r... | 1 | [
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10. It is required to fence a rectangular area adjacent to a wall. The fence should have a length of \(60 \mathrm{~m}\). What should be the length and width of this area so that its area is the largest? | First, I need to visualize the problem. There's a rectangular area adjacent to a wall, so one side of the rectangle doesn't require fencing.
Let’s denote the length parallel to the wall as \( L \) and the width perpendicular to the wall as \( W \).
The total length of the fence needed is the sum of the length and twi... | 0 | [
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"31. Given that $\\alpha$ is an acute angle satisfying\n$$\n\\sqrt{369-360 \\cos \\alpha}+\\sqrt{544(...TRUNCATED) | "Alright, so I have this problem here where I need to find the value of 40 tan α, given that α is (...TRUNCATED) | 1 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [71486,11,773,358,614,419,3491,1588,1380,358,1184,311,1477,279,897,315,220,19,15,14197,19043,11,2661(...TRUNCATED) | [[0.00004739686846733093,0.000037049874663352966,0.00005853300172020681,0.00005414383485913277,0.000(...TRUNCATED) |
"## Task 7 - V00607 = V00503\n\nTo an imagined number, 16 is added, then multiplied by 7, followed b(...TRUNCATED) | "First, I'll represent the imagined number as \\( x \\).\n\nWhen 16 is added to \\( x \\), the expre(...TRUNCATED) | 1 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [5338,11,358,3278,4009,279,34606,1372,438,17767,856,1124,3593,4498,220,16,21,374,3694,311,17767,856,(...TRUNCATED) | [[0.0007344484329223633,0.0005984902381896973,0.0005827744607813656,0.0006236657500267029,0.00079512(...TRUNCATED) |
"IMO 1974 Problem A1 Three players play the following game. There are three cards each with a differ(...TRUNCATED) | "Alright, so I’ve got this problem from IMO 1974, Problem A1. It involves three players who each s(...TRUNCATED) | 0 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [71486,11,773,358,3982,2684,419,3491,504,84071,220,16,24,22,19,11,22079,362,16,13,1084,17601,2326,42(...TRUNCATED) | [[0.000030172988772392273,0.00002446584403514862,0.000013860563740308862,0.000014015240594744682,0.0(...TRUNCATED) |
9.1. Solve the equation:
$$
\left(x^{2}-20\right)^{2}+\left(x^{2}-19\right)^{2}=2019
$$ | "Okay, so I've got this equation to solve: \\((x^2 - 20)^2 + (x^2 - 19)^2 = 2019\\). Hmm, let me try(...TRUNCATED) | 0 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [32313,11,773,358,3003,2684,419,23606,311,11625,25,1124,1188,87,61,17,481,220,17,15,29776,17,488,320(...TRUNCATED) | [[0.0000957585871219635,0.00017478317022323608,0.00008601943409303203,0.00004909397102892399,0.00003(...TRUNCATED) |
"19. (CAN 5) The triangular array $\\left(a_{n, k}\\right)$ of numbers is given by $a_{n, 1}=1 / n$,(...TRUNCATED) | "Okay, so I have this problem here about a triangular array of numbers, and I need to find the harmo(...TRUNCATED) | 1 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [32313,11,773,358,614,419,3491,1588,911,264,65494,1334,315,5109,11,323,358,1184,311,1477,279,81241,3(...TRUNCATED) | [[0.00006004050374031067,0.000033866846933960915,0.000045243650674819946,0.000044899992644786835,0.0(...TRUNCATED) |
". For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible va(...TRUNCATED) | "Alright, so I'm trying to solve this problem where I need to find the largest possible value of the(...TRUNCATED) | 1 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [71486,11,773,358,2776,4460,311,11625,419,3491,1380,358,1184,311,1477,279,7772,3204,897,315,279,7493(...TRUNCATED) | [[0.00009372830390930176,0.000038771890103816986,0.00003559018296073191,0.00003343820571899414,0.000(...TRUNCATED) |
"In equilateral triangle $ABC$, the midpoint of $\\overline{BC}$ is $M$. If the circumcircle of tria(...TRUNCATED) | "Okay, so I have this problem about an equilateral triangle ABC. Let me first try to visualize it. S(...TRUNCATED) | 1 | [151646,198,27,91,318,4906,91,29,8948,198,2610,525,1207,16948,11,3465,553,54364,14817,13,1446,525,26(...TRUNCATED) | [32313,11,773,358,614,419,3491,911,458,3232,42939,21495,19360,13,6771,752,1156,1430,311,50087,432,13(...TRUNCATED) | [[0.00015847664326429367,0.000028164591640233994,0.000014878732145007234,0.00002707267412915826,0.00(...TRUNCATED) |
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