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) | [2.734375,5.65625,6.21875,2.609375,4.53125,2.984375,2.984375,5.6875,0.5703125,2.78125,-0.04907226562(...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) | [-5.03125,-4.125,-4.21875,-3.265625,-4.625,-4.9375,-4.4375,-3.296875,-2.71875,-3.0625,-5.59375,-3.81(...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) | [-4.53125,-4.71875,-5.0625,-4.5625,-4.75,-5.15625,-4.59375,-4.40625,-4.46875,-4.625,-4.59375,-4.5,-4(...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) | [0.59375,-0.2412109375,2.109375,1.625,0.98828125,1.2265625,0.57421875,1.2578125,-0.294921875,1.70312(...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) | [-0.88671875,-1.984375,1.25,-2.140625,-1.640625,-2.09375,-2.15625,-2.15625,-1.9375,-2.109375,-2.4062(...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) | [0.1484375,2.96875,-0.458984375,2.515625,-0.337890625,2.1875,2.296875,1.5390625,0.419921875,0.185546(...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) | [-1.8359375,0.1796875,-5.0,-3.15625,-6.625,-3.8125,-0.33203125,-5.40625,-1.9453125,-2.96875,-5.59375(...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) | [-4.25,-4.9375,-4.0625,-1.296875,-4.375,-4.71875,-4.46875,-2.546875,-4.53125,-5.03125,-1.8203125,-4.(...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) | [2.640625,3.828125,3.0625,0.97265625,3.0625,-0.90234375,1.78125,0.2109375,1.1796875,3.390625,3.35937(...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) | [-3.578125,-0.84765625,-5.5,-3.171875,-3.234375,-5.625,-5.15625,-5.28125,-4.96875,-3.265625,-3.21875(...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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