messages listlengths 2 2 | answer stringlengths 0 1.22k | has_detailed_solution bool 2
classes |
|---|---|---|
[
{
"role": "user",
"content": "The operation $\\otimes$ is defined for all nonzero numbers by $a \\otimes b = \\frac{a^{2}}{b}$. Determine $[(1 \\otimes 2) \\otimes 3] - [1 \\otimes (2 \\otimes 3)]$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to figure out this proble... | -\frac{2}{3} | true |
[
{
"role": "user",
"content": "Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?"
},
{
"role": "assi... | 26 | true |
[
{
"role": "user",
"content": "Let $P(x)$ be a polynomial of degree $3n$ such that\n\\begin{align*} P(0) = P(3) = \\dots = P(3n) &= 2, \\\\ P(1) = P(4) = \\dots = P(3n+1-2) &= 1, \\\\ P(2) = P(5) = \\dots = P(3n+2-2) &= 0. \\end{align*}\nAlso, $P(3n+1) = 730$. Determine $n$."
},
{
"role": "assistant"... | 1 | true |
[
{
"role": "user",
"content": "Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to solve this problem. The function is given as f(x) = a x² - √2, and we need t... | \frac{\sqrt{2}}{2} | true |
[
{
"role": "user",
"content": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students... | 28 | true |
[
{
"role": "user",
"content": "If $991+993+995+997+999=5000-N$, then $N=$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the value of N in the equation 991 + 993 + 995 + 997 + 999 = 5000 - N. Hmm, let me think. First, maybe I should calculate the sum on the left side and ... | 25 | true |
[
{
"role": "user",
"content": "The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser?"
... | 7 | true |
[
{
"role": "user",
"content": "Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ a... | \frac{5}{3} | true |
[
{
"role": "user",
"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so Judy had 35 hits in tota... | 80\% | true |
[
{
"role": "user",
"content": "The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out the equation of the graph G' which ... | 10^{-x} | true |
[
{
"role": "user",
"content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so ... | 7 | true |
[
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flow... | \frac{1}{5} | true |
[
{
"role": "user",
"content": "What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\\cos 40^\\circ,\\sin 40^\\circ)$, $(\\cos 60^\\circ,\\sin 60^\\circ)$, and $(\\cos t^\\circ,\\sin t^\\circ)$ is isosceles?"
},
{
"rol... | 380 | true |
[
{
"role": "user",
"content": "In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at p... | 20 | true |
[
{
"role": "user",
"content": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the value of x for which the first three terms of an arithmetic progressio... | 0 | true |
[
{
"role": "user",
"content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of ... | \frac{9}{10} | true |
[
{
"role": "user",
"content": "An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, ... | 17 | true |
[
{
"role": "user",
"content": "The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the value or values of k such that the three points (2, -3), (4, 3), and (5, k/2) lie on... | 12 | true |
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