prompt stringlengths 20 1.73k | answers sequencelengths 1.02k 1.02k | rewards sequencelengths 1.02k 1.02k | label sequencelengths 1.02k 1.02k |
|---|---|---|---|
"Convert the point $(0,3)$ in rectangular coordinates to polar coordinates. Enter your answer in th(...TRUNCATED) | [" The rectangular coordinates $(x,y)$ can be converted to polar coordinates $(r,\\theta)$ using the(...TRUNCATED) | [2.609375,2.015625,2.21875,1.8359375,1.171875,2.59375,2.890625,1.1640625,0.97265625,0.0830078125,2.7(...TRUNCATED) | [1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1(...TRUNCATED) |
"Define\n\\[p = \\sum_{k = 1}^\\infty \\frac{1}{k^2} \\quad \\text{and} \\quad q = \\sum_{k = 1}^\\i(...TRUNCATED) | [" We can rewrite the sum as\n\\begin{align*}\n\\sum_{j = 1}^\\infty \\sum_{k = 1}^\\infty \\frac{1}(...TRUNCATED) | [1.6640625,0.6796875,0.96484375,-0.58984375,-5.0625,-4.15625,-0.47265625,-1.875,0.95703125,-1.09375,(...TRUNCATED) | [0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"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) | [" To solve this problem, we need to substitute $x$ with $-2$, $-1$, and $0$ in the function $f(x)$ (...TRUNCATED) | [2.6875,3.21875,1.9609375,3.03125,2.234375,2.53125,3.109375,3.09375,2.1875,3.140625,2.96875,2.125,1.(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1(...TRUNCATED) |
How many positive whole-number divisors does 196 have? | [" To find the number of divisors of 196, we first need to find its prime factorization. \n\nThe pri(...TRUNCATED) | [1.953125,3.71875,3.015625,3.03125,2.359375,2.75,1.2421875,1.796875,0.8203125,1.4140625,0.83203125,2(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"The results of a cross-country team's training run are graphed below. Which student has the greates(...TRUNCATED) | [" To find out which student has the greatest average speed, we need to calculate the average speed (...TRUNCATED) | [1.78125,0.98828125,0.82421875,0.55078125,0.62109375,-1.1953125,1.6640625,-0.181640625,1.3203125,0.3(...TRUNCATED) | [0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,1,0,0,0,0,1,0,0,1,1,1,0,0(...TRUNCATED) |
"A regular hexagon can be divided into six equilateral triangles. If the perimeter of one of the tri(...TRUNCATED) | [" The perimeter of the hexagon is made up of six times the base of one of the triangles, because al(...TRUNCATED) | [1.640625,1.6015625,-0.6953125,1.8359375,2.3125,1.4375,1.953125,1.7890625,1.75,1.8515625,2.234375,0.(...TRUNCATED) | [1,0,0,0,1,0,0,1,1,1,1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0,1,1,1,0,1,1,0,1,0,0,0,0,1,1(...TRUNCATED) |
"What is the smallest positive perfect cube that can be written as the sum of three consecutive inte(...TRUNCATED) | [" Let's denote the three consecutive integers as $n$, $n+1$ and $n+2$. \n\nThe sum of these three i(...TRUNCATED) | [-2.90625,1.8046875,-4.125,-0.423828125,-5.3125,-3.203125,0.328125,-4.1875,-0.09130859375,-4.0,-4.68(...TRUNCATED) | [0,1,0,0,0,0,1,0,1,0,0,1,1,1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,1,1,1,1,0,1,1,0,1,0,1,1,1,1,1,0,1,1(...TRUNCATED) |
"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) | [" Let's denote the direction vectors of the two lines as $\\textbf{d}_1 = (2, 3, -1)$ and $\\textbf(...TRUNCATED) | [-2.546875,-1.421875,-3.6875,-1.1171875,-3.0625,-1.4140625,-3.546875,-0.7578125,-2.96875,-3.578125,-(...TRUNCATED) | [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0(...TRUNCATED) |
"What is the distance, in units, between the points $(2, -6)$ and $(-4, 3)$? Express your answer in (...TRUNCATED) | [" The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a plane can be calculated using (...TRUNCATED) | [2.125,1.890625,1.8984375,2.953125,2.5625,3.234375,2.0625,1.6015625,2.390625,2.34375,2.125,1.9921875(...TRUNCATED) | [1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"The expression $2\\cdot 3 \\cdot 4\\cdot 5+1$ is equal to 121, since multiplication is carried out (...TRUNCATED) | [" To solve this problem, we need to consider all possible ways to insert parentheses into the expre(...TRUNCATED) | [-3.125,0.052734375,-4.625,-1.5703125,-2.921875,-5.625,-1.5546875,-3.453125,-3.125,-2.234375,1.09375(...TRUNCATED) | [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
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