prompt_text stringclasses 500 values | prompt_ids listlengths 10 864 | decode_ids listlengths 64 64 | prompt_pattern listlengths 27 27 | decode_pattern listlengths 27 27 |
|---|---|---|---|---|
Convert the point $(0,3)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | [
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Define
\[p = \sum_{k = 1}^\infty \frac{1}{k^2} \quad \text{and} \quad q = \sum_{k = 1}^\infty \frac{1}{k^3}.\]Find a way to write
\[\sum_{j = 1}^\infty \sum_{k = 1}^\infty \frac{1}{(j + k)^3}\]in terms of $p$ and $q.$ | [
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"If $f(x) = \\frac{3x-2}{x-2}$, what is the value of $f(-2) +f(-1)+f(0)$? Express your answer as a c(...TRUNCATED) | [100000,1847,363,69,7,87,8,403,357,1122,90,18,87,12,17,1061,87,12,17,3001,856,317,254,1437,280,363,6(...TRUNCATED) | [185,185,185,23853,207,21,13,16,19,185,2640,317,254,1437,280,363,69,7,18,12,18,8226,69,7,18,1026,30,(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[14,34,42,56,8,45,36,57],[14,40,49,16,46,34,7,1],[5,24,0,34,31,10,11,4],[4(...TRUNCATED) | [[[2,47,14,40,59,43,15,44],[47,14,2,40,34,15,59,44],[47,2,34,54,15,52,61,14],[1,42,62,34,27,58,35,32(...TRUNCATED) |
How many positive whole-number divisors does 196 have? | [
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207,
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21,
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] | [185,185,185,1862,1122,90,16,11,17,11,19,11,23,11,24,11,16,21,11,16,17,11,16,3,185,185,185,2054,481,(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[53,56,8,1,42,34,14,54],[60,21,38,58,14,26,20,10],[38,19,58,10,23,20,60,55(...TRUNCATED) | [[[14,47,2,13,40,44,51,15],[14,47,2,40,13,51,44,15],[47,2,14,34,15,61,1,43],[16,1,53,49,34,7,46,14],(...TRUNCATED) |
"The results of a cross-country team's training run are graphed below. Which student has the greates(...TRUNCATED) | [100000,549,2472,280,245,4234,12,14821,2557,6,82,4188,1409,418,32591,874,2877,13,11692,6710,643,254,(...TRUNCATED) | [185,185,185,12423,6332,185,185,12423,1913,185,774,254,8499,2330,11,254,4583,317,11096,457,254,3973,(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[14,34,8,6,20,49,57,12],[63,59,23,35,50,41,62,1],[21,56,9,50,63,14,40,39],(...TRUNCATED) | [[[47,2,59,37,54,62,22,11],[47,2,59,37,54,62,22,11],[47,2,37,54,59,62,22,35],[20,50,42,41,62,1,57,4](...TRUNCATED) |
"A regular hexagon can be divided into six equilateral triangles. If the perimeter of one of the tri(...TRUNCATED) | [100000,32,4252,22161,5995,481,330,13429,881,4484,1517,64644,38751,13,1273,254,55389,280,634,280,254(...TRUNCATED) | [185,185,549,4252,22161,5995,317,10237,11,185,1567,317,245,4252,22161,5995,11,185,32,4252,22161,5995(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[37,5,34,33,24,25,31,42],[33,21,37,19,57,34,20,60],[33,60,24,34,37,26,4,5](...TRUNCATED) | [[[2,47,14,13,44,25,22,18],[2,47,14,13,44,25,3,18],[8,6,34,14,12,57,42,49],[63,21,19,23,20,59,60,45](...TRUNCATED) |
"What is the smallest positive perfect cube that can be written as the sum of three consecutive inte(...TRUNCATED) | [
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"The set of points $(x,y,z)$ that satisfy\n\\[2x = 3y = -z\\]is a line.\n\nThe set of points $(x,y,z(...TRUNCATED) | [100000,549,845,280,3487,3309,87,11,88,11,89,1026,344,12276,185,6844,17,87,403,207,18,88,403,570,89,(...TRUNCATED) | [185,100001,100000,16237,254,845,280,521,3039,280,363,87,3,279,254,11501,12669,15,11,17,59,1804,8826(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[14,34,8,6,49,20,57,42],[63,34,33,59,50,60,23,41],[56,63,21,9,20,59,14,22](...TRUNCATED) | [[[47,2,18,58,22,3,59,62],[43,34,23,42,50,4,45,1],[14,34,42,1,63,52,61,59],[42,41,15,20,38,50,57,29](...TRUNCATED) |
"What is the distance, in units, between the points $(2, -6)$ and $(-4, 3)$? Express your answer in (...TRUNCATED) | [100000,2640,317,254,5013,11,279,8650,11,1439,254,3487,3309,17,11,570,21,1026,285,65026,19,11,207,18(...TRUNCATED) | [185,185,185,40,6,313,803,3507,276,830,331,437,327,245,1477,1132,285,953,481,6,83,1751,276,752,688,3(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[8,34,53,42,1,20,14,56],[4,36,14,30,34,20,39,5],[12,14,52,8,40,27,18,21],[(...TRUNCATED) | [[[2,47,14,43,15,22,44,58],[2,47,14,43,15,44,62,40],[47,2,14,34,15,54,61,52],[48,5,42,22,24,57,8,25](...TRUNCATED) |
"The expression $2\\cdot 3 \\cdot 4\\cdot 5+1$ is equal to 121, since multiplication is carried out (...TRUNCATED) | [100000,549,5761,363,17,59,3560,207,18,357,3560,207,19,59,3560,207,20,10,16,3,317,5929,276,207,16,17(...TRUNCATED) | [100001,100000,11083,6787,285,9335,276,769,4921,2653,9122,331,17678,323,91752,13,62667,6,82,2508,491(...TRUNCATED) | [[[14,34,2,42,47,1,46,40],[14,34,8,6,20,49,57,12],[63,23,59,50,55,4,37,34],[40,49,16,14,7,46,34,1],[(...TRUNCATED) | [[[43,50,27,29,49,6,46,0],[14,34,2,47,42,52,8,53],[42,19,38,10,20,34,31,37],[38,58,27,0,43,10,37,34](...TRUNCATED) |
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