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\nThe conversion formulas from rectangular coordinates $(x, y)$ to polar coordinates $(r, \\theta)$ are $r = \\sqrt{x^2 + y^2}$ and $\\theta = \\tan^{-1}\\left(\\frac{y}{x}\\right)$, where $r > 0$ and $0 \\le \\theta < 2\\pi$... |
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: We're given two series, $p$ and $q$, defined as $p = \\sum_{k = 1}^\\infty \\frac{1}{k^2}$ and $q = \\sum_{k = 1}^\\infty \\frac{1}{k^3}$, and we need to express the series $\\sum_{j = 1}^\\infty \\sum_{k = 1}^\\infty \\frac{1}{(j + k)^3}$ in terms of $p$ and $q$.\n## Step 2: To simplify the given ser... |
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)\nWe need to substitute -2 into the function f(x) = (3x-2)/(x-2).\nf(-2) = (3(-2)-2)/(-2-2) = (-6-2)/(-4) = -8/(-4) = 2\n\n## Step 2: Evaluate f(-1)\nWe need to substitute -1 into the function f(x) = (3x-2)/(x-2).\nf(-1) = (3(-1)-2)/(-1-2) = (-3-2)/(-3) = -5/(-3) = 5/3\n\n## Step 3: Evalua... |
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: Prime factorize 196\nTo find the number of positive whole-number divisors of 196, we first need to find its prime factorization. 196 can be factored into 2^2 * 7^2.\n\n## Step 2: Apply the divisor formula\nFor a number in the form of p^a * q^b, where p and q are prime numbers, the total number of diviso... |
The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [asy]
for ( int i = 1; i <= 7; ++i )
{
draw((i,0)--(i,6));
}
for ( int i = 1; i <= 5; ++i )
{
draw((0,i)--(8,i));
}
draw((-0.5,0)--(8,0), linewidth(1));
draw((0,-0.5)--(0,6), linewidth(1));
label("$O$"... | Evelyn covered more distance in less time than Briana, Debra and Angela, so her average speed is greater than any of their average speeds. Evelyn went almost as far as Carla in less than half the time that it took Carla, so Evelyn's average speed is also greater than Carla's. Therefore, $\boxed{\text{Evelyn}}$ is our a... | \text{Evelyn} | Algebra | 2 | test/algebra/1349.json | [
"## Step 1: To find the average speed of each student, we need to calculate the distance covered and the time taken by each student.\nThe distance covered and time taken by each student can be determined from the graph.\n\n## Step 2: For Evelyn, the distance covered is 4.5 and the time taken is 4, so the average sp... |
"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: Understand the relationship between the perimeter of a triangle and the hexagon\nSince (...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\nWe are looking for the smallest positive perfect cube that can (...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: Express the first line in vector form\nWe start by expressing the first line in vector (...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: Identify the coordinates of the two points\nThe coordinates of the first point are $(2,(...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\nTo find the total number of values that can be obtained from th(...TRUNCATED) |
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