problem stringlengths 20 1.73k | solution stringlengths 45 3.36k | answer stringlengths 1 53 | subject stringclasses 7
values | level int64 1 5 | unique_id stringlengths 20 40 | responses sequencelengths 32 32 |
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Convert the point $(0,3)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | We have that $r = \sqrt{0^2 + 3^2} = 3.$ Also, if we draw the line connecting the origin and $(0,3),$ this line makes an angle of $\frac{\pi}{2}$ with the positive $x$-axis.
[asy]
unitsize(0.8 cm);
draw((-0.5,0)--(3.5,0));
draw((0,-0.5)--(0,3.5));
draw(arc((0,0),3,0,90),red,Arrow(6));
dot((0,3), red);
label("$(0,3)... | \left( 3, \frac{\pi}{2} \right) | Precalculus | 2 | test/precalculus/807.json | [
"## Step 1: Recall the formulas for converting rectangular coordinates to polar coordinates.\nTo convert a point from rectangular coordinates $(x,y)$ to polar coordinates $(r,\\theta),$ we use the following formulas: $r = \\sqrt{x^2 + y^2}$ for the radial coordinate and $\\theta = \\tan^{-1}\\left(\\frac{y}{x}\\rig... |
Define
\[p = \sum_{k = 1}^\infty \frac{1}{k^2} \quad \text{and} \quad q = \sum_{k = 1}^\infty \frac{1}{k^3}.\]Find a way to write
\[\sum_{j = 1}^\infty \sum_{k = 1}^\infty \frac{1}{(j + k)^3}\]in terms of $p$ and $q.$ | We count the number of times $\frac{1}{n^3}$ appears in the sum
\[\sum_{j = 1}^\infty \sum_{k = 1}^\infty \frac{1}{(j + k)^3},\]where $n$ is a fixed positive integer. (In other words, we are conditioning the sum on $j + k$.) We get a term of $\frac{1}{n^3}$ each time $j + k = n.$ The pairs $(j,k)$ that work are $(1,... | p - q | Intermediate Algebra | 5 | test/intermediate_algebra/1994.json | [
"## Step 1: Express the given series as a sum over $j$ and $k$\nWe are given the series $\\sum_{j = 1}^\\infty \\sum_{k = 1}^\\infty \\frac{1}{(j + k)^3}.$ To rewrite this series in terms of $p$ and $q,$ let's first try to simplify the expression by making a substitution.\n\n## Step 2: Make the substitution $m = j ... |
If $f(x) = \frac{3x-2}{x-2}$, what is the value of $f(-2) +f(-1)+f(0)$? Express your answer as a common fraction. | $f(-2)+f(-1)+f(0)=\frac{3(-2)-2}{-2-2}+\frac{3(-1)-2}{-1-2}+\frac{3(0)-2}{0-2}=\frac{-8}{-4}+\frac{-5}{-3}+\frac{-2}{-2}=2+\frac{5}{3}+1=\boxed{\frac{14}{3}}$ | \frac{14}{3} | Algebra | 3 | test/algebra/2584.json | [
"## Step 1: Evaluate $f(-2)\nFirst, let's evaluate $f(-2)$. Plug $x=-2$ into the function $f(x) = \\frac{3x-2}{x-2}$.\n$f(-2) = \\frac{3(-2) - 2}{-2 - 2} = \\frac{-6 - 2}{-4} = \\frac{-8}{-4} = 2$\n\n## Step 2: Evaluate $f(-1)\nNow, let's evaluate $f(-1)$. Plug $x=-1$ into the function $f(x) = \\frac{3x-2}{x-2}$.\n... |
How many positive whole-number divisors does 196 have? | First prime factorize $196=2^2\cdot7^2$. The prime factorization of any divisor of 196 cannot include any primes other than 2 and 7. We are free to choose either 0, 1, or 2 as the exponent of 2 in the prime factorization of a divisor of 196. Similarly, we may choose 0, 1, or 2 as the exponent of 7. In total, there ... | 9 | Number Theory | 3 | test/number_theory/572.json | [
"## Step 1: Find the prime factorization of 196\nTo find the number of positive whole-number divisors of 196, we first need to find its prime factorization. 196 can be factored as $2^2 \\cdot 7^2$.\n\n## Step 2: Use the formula to find the number of divisors\nThe formula for the number of divisors of a number $n = ... |
"The results of a cross-country team's training run are graphed below. Which student has the greates(...TRUNCATED) | "Evelyn covered more distance in less time than Briana, Debra and Angela, so her average speed is gr(...TRUNCATED) | \text{Evelyn} | Algebra | 2 | test/algebra/1349.json | ["## Step 1: To find the student with the greatest average speed, we need to calculate the average (...TRUNCATED) |
"A regular hexagon can be divided into six equilateral triangles. If the perimeter of one of the tri(...TRUNCATED) | "The side length of the hexagon is equal to the side length of one of the equilateral triangles. Si(...TRUNCATED) | 42 | Prealgebra | 2 | test/prealgebra/1622.json | ["## Step 1: Since the hexagon can be divided into six equilateral triangles, the perimeter of the h(...TRUNCATED) |
"What is the smallest positive perfect cube that can be written as the sum of three consecutive inte(...TRUNCATED) | "The sum of three consecutive integers takes the form $(k-1)+(k)+(k+1)=3k$ and hence is a multiple o(...TRUNCATED) | 27 | Number Theory | 3 | test/number_theory/515.json | ["## Step 1: Understand the problem\nTo find the smallest positive perfect cube that can be written (...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) | "For the first line, let $t = 2x = 3y = -z.$ Then\n\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatri(...TRUNCATED) | 90^\circ | Precalculus | 4 | test/precalculus/927.json | ["## Step 1: Identify the direction vectors of the two lines.\nTo find the direction vectors, we fir(...TRUNCATED) |
"What is the distance, in units, between the points $(2, -6)$ and $(-4, 3)$? Express your answer in (...TRUNCATED) | "We use the distance formula: \\begin{align*}\n\\sqrt{(2 - (-4))^2 + ((-6) - 3)^2} &= \\sqrt{6^2 + (...TRUNCATED) | 3\sqrt{13} | Algebra | 3 | test/algebra/2036.json | ["## Step 1: To find the distance between two points, we can use the distance formula.\nThe distance(...TRUNCATED) |
"The expression $2\\cdot 3 \\cdot 4\\cdot 5+1$ is equal to 121, since multiplication is carried out (...TRUNCATED) | "By the associative property of multiplication, it doesn't help to insert parentheses that specify t(...TRUNCATED) | 4 | Prealgebra | 5 | test/prealgebra/1139.json | ["## Step 1: Understand the problem and the allowed operations\nWe need to find the total number of (...TRUNCATED) |
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