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"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) | [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) | [0.622459352016449,0.8354835510253906,0.8670357465744019,0.22270014882087708,0.9626730680465698,0.53(...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) | [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) | [0.004070138093084097,0.005220125894993544,0.0031726830638945103,0.0031726830638945103,0.00116951018(...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) | [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) | [0.01798621006309986,0.03308597952127457,0.01798621006309986,0.022977370768785477,0.0124316513538360(...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) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0.8354835510253906,0.7310585379600525,0.9859364032745361,0.8354835510253906,0.851952850818634,0.924(...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) | [0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,1,0,0,0,0,1,0,0,1,1,1,0,0(...TRUNCATED) | [0.9525741338729858,0.7549149990081787,0.7310585379600525,0.7549149990081787,0.7772998809814453,0.40(...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) | [1,0,0,0,1,0,0,1,1,1,1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0,1,1,1,0,1,1,0,1,0,0,0,0,1,1(...TRUNCATED) | [0.6791787147521973,0.7057850360870361,0.29421496391296387,0.7310585379600525,0.43782350420951843,0.(...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) | [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) | [0.10669060051441193,0.7057850360870361,0.01243165135383606,0.08509904146194458,0.001032231026329100(...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) | [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) | [0.02931223064661026,0.01590639166533947,0.022977370768785477,0.14804719388484955,0.0109869437292218(...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) | [1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0.9324532747268677,0.957912266254425,0.8807970285415649,0.8807970285415649,0.9626730680465698,0.562(...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) | [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) | [0.037326883524656296,0.01798621006309986,0.006692850962281227,0.07585817575454712,0.033085979521274(...TRUNCATED) |
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