question string | ground_truth string | model_outputs list | extracted_answers list | is_correct list |
|---|---|---|---|---|
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | 204 | [
"To find the time Aya takes for the walk at $s+\\frac{1}{2}$ kilometers per hour, we first need to find the values of $s$ and $t$.\n\nGiven that the walk at $s$ kilometers per hour takes 4 hours, we can write the equation:\n\n$$\\frac{9}{s} + \\frac{t}{60} = 4$$\n\nGiven that the walk at $s+2$ kilometers per hour t... | [
"466",
"215",
"214",
"336"
] | [
false,
false,
false,
false
] |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$.
Answ... | 113 | [
"To solve this problem, we will use the power of a point theorem and Stewart's theorem.\n\nLet's denote the points where the tangents from A to $\\omega$ intersect the circle as $B'$ and $C'$, and let $x = BP$, $y = CP$, $z = AP$, $r = AB' = AC'$.\n\nBy the power of a point theorem, we have $BP \\cdot PC = BA \\cdo... | [
"119",
"6",
"11",
""
] | [
false,
false,
false,
false
] |
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | 371 | [
"To find the probability that the octagon can be rotated so that all of the blue vertices end up at positions where there were originally red vertices, we need to consider the possible colorings of the octagon.\n\nLet's assume the octagon has 8 vertices, and we'll use the variable $B$ to represent a blue vertex and... | [
"147",
"",
"322",
"23"
] | [
false,
false,
false,
false
] |
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$. Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\]
Answer the query and place the final integer inside \boxed{} in your response. For example \boxed{42}. | 385 | [
"To find the number of intersections of the graphs of $y=4 g(f(\\sin (2 \\pi x)))$ and $x=4 g(f(\\cos (3 \\pi y)))$, we can analyze the given functions.\n\nFirst, let's simplify $f(x)$ and $g(x)$. \n\n$f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$\n\nWe know that $|x| = x$ if $x \\geq 0$ and $|x| = -x$... | [
"8",
"16",
"52",
""
] | [
false,
false,
false,
false
] |
Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.
Answer the query and place the final integer inside \boxed{} in your response. For example \boxed{42}. | 110 | [
"To solve this problem, we first need to find the least prime number $p$ for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$.\n\nWe note that $n^{4}+1$ can be factored as $(n^{2}+1)^{2}-n^{2}$. Let $x=n^{2}$. Then we want to find the smallest prime $p$ such that there exists a ... | [
"5",
"",
"",
""
] | [
false,
false,
false,
false
] |
"Let $ABCD$ be a tetrahedron such that $AB=CD= \\sqrt{41}$, $AC=BD= \\sqrt{80}$, and $BC=AD= \\sqrt{(...TRUNCATED) | 104 | ["To solve this problem, we first need to find the point I inside the tetrahedron. \n\nThe distance (...TRUNCATED) | [
"514",
"",
"",
""
] | [
false,
false,
false,
false
] |
"Let $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$. Let $r$ (...TRUNCATED) | 721 | ["To find the smallest sphere that can contain each of the rectangular boxes in $\\mathcal{B}$, we n(...TRUNCATED) | [
"95",
"37",
"",
""
] | [
false,
false,
false,
false
] |
"There exist real numbers $x$ and $y$, both greater than 1, such that $\\log_x\\left(y^x\\right)=\\l(...TRUNCATED) | 25 | ["Given that $\\log_x\\left(y^x\\right)=\\log_y\\left(x^{4y}\\right)=10$, we can rewrite the first e(...TRUNCATED) | [
"",
"7",
"",
"6.25"
] | [
false,
false,
false,
false
] |
"Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take tur(...TRUNCATED) | 809 | ["To find the number of positive integers n for which Bob has a winning strategy, we need to conside(...TRUNCATED) | [
"1216",
"808",
"1216",
""
] | [
false,
false,
false,
false
] |
"Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers(...TRUNCATED) | 116 | ["To calculate the probability of Jen winning the grand prize given that she won a prize, we first n(...TRUNCATED) | [
"116",
"116",
"115",
"116"
] | [
true,
true,
false,
true
] |
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