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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) | [0.9149009585380554,0.851952850818634,0.8807970285415649,0.562176525592804,0.8807970285415649,0.9241(...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) | [0.531209409236908,0.06754669547080994,0.4687906503677368,0.2018132209777832,0.01590639166533947,0.0(...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) | [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) | [0.8175744414329529,0.7549149990081787,0.8807970285415649,0.7981867790222168,0.7772998809814453,0.86(...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) | [0.10669060051441193,0.8354835510253906,0.8807970285415649,0.11920291930437088,0.24508501589298248,0(...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) | [0.7549149990081787,0.562176525592804,0.5926666259765625,0.622459352016449,0.531209409236908,0.65135(...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) | [0.5,0.43782350420951843,0.14804719388484955,0.3208213150501251,0.8175744414329529,0.348645150661468(...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) | [0.11920291930437088,0.5,0.005911068990826607,0.042087726294994354,0.007577240467071533,0.0059110689(...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) | [0.11920291930437088,0.14804719388484955,0.06008664518594742,0.2018132209777832,0.037326883524656296(...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) | [0.8670357465744019,0.8354835510253906,0.8933094143867493,0.8670357465744019,0.8933094143867493,0.92(...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) | [0.04742587357759476,0.6513549089431763,0.014063626527786255,0.08509904146194458,0.02931223064661026(...TRUNCATED) |
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