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) | [0.94140625,3.8125,3.75,0.8359375,3.671875,1.2421875,1.2421875,4.40625,0.2490234375,2.921875,0.17871(...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) | [-4.1875,-4.46875,-4.84375,-6.15625,-6.1875,-6.03125,-4.90625,-5.6875,-4.0,-4.5,-6.65625,-5.0,-5.187(...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) | [-5.375,-4.78125,-4.65625,-5.15625,-5.65625,-4.625,-5.28125,-5.4375,-5.4375,-4.96875,-4.96875,-5.812(...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) | [1.5234375,-1.90625,1.7109375,3.0,2.25,3.828125,0.365234375,1.8671875,-0.1884765625,1.75,1.1796875,0(...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) | [1.75,0.050048828125,0.59765625,-1.5234375,-1.1953125,-1.46875,-0.83984375,-2.4375,-1.5546875,0.2753(...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) | [0.93359375,1.765625,-0.466796875,1.7109375,-0.45703125,2.53125,1.3515625,1.515625,2.921875,0.535156(...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) | [-2.484375,0.484375,-4.875,-3.59375,-6.28125,-5.4375,-1.3984375,-6.625,-3.203125,-4.34375,-5.875,0.1(...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) | [-5.75,-6.21875,-4.125,-2.375,-5.15625,-4.46875,-5.0,-2.65625,-5.84375,-5.84375,-1.71875,-2.265625,-(...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) | [1.1171875,1.2421875,1.1171875,0.28515625,1.4453125,-1.28125,0.59765625,-0.333984375,0.50390625,1.23(...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) | [-4.0625,-4.0625,-6.75,-3.765625,-4.21875,-5.78125,-5.34375,-5.8125,-5.46875,-4.03125,-3.03125,-4.65(...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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