prompt list | label string |
|---|---|
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$... | 73 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A list of positive integers has the following properties: $\\bullet$ The sum of the items in the list is $30$ . $\\bullet$ The unique mode of the list is $9$ . $\\bullet$ The medi... | 236 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$ , and the sum of the three numbers ... | 45 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: \\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\] \\[\\log_2\\left({y \\over xz}\... | 33 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCDEF$ be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments $\\overline{AB}$ , $\\overline{... | 80 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$ . B... | 55 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$ , the resulting number is divisible by $7$ . Let $Q$ and $R$ be the ... | 699 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Torus $T$ is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle (so like a donut... | 127 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in a $5 \times 5$ grid such that\n- each ... | 902 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle$ $ABC$ have incenter $I$ and circumcenter $O$ with $\\overline{IA} \\perp \\overline{OI}$ , circumradius $13$ , and inradius $6$ . Find $AB \\cdot AC$ .",
"rol... | 468 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and \\[a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6,000,000.\\]",
"role": "use... | 601 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $O(0,0),A(\\tfrac{1}{2},0),$ and $B(0,\\tfrac{\\sqrt{3}}{2})$ be points in the coordinate plane. Let $\\mathcal{F}$ be the family of segments $\\overline{PQ}$ of unit length l... | 23 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when \\[\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\] is divided by 1000.",
"role": "user"
}
] | 321 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $b \\geq 2$ be an integer. Call a positive integer $n$ $b\\textit{-eautiful}$ if it has exactly two digits when expressed in base $b$ , and these two digits sum to $\\sqrt{n}$... | 211 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of rectangles that can be formed inside a fixed regular dodecagon ( $12$ -gon) where each side of the rectangle lies on either a side or a diagonal of the dodecago... | 315 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Every morning, Aya does a $9$ kilometer walk, and then finishes at the coffee shop. One day, she walks at $s$ kilometers per hour, and the walk takes $4$ hours, including $t$ minu... | 204 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Real numbers $x$ and $y$ with $x,y>1$ satisfy $\\log_x(y^x)=\\log_y(x^{4y})=10.$ What is the value of $xy$ ?",
"role": "user"
}
] | 25 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes $1$ token or $4$ tok... | 809 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers w... | 116 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $... | 104 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\\times 8$ grid. Find the number of such paths that change dir... | 294 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the largest possible real part of \\[(75+117i)z+\\frac{96+144i}{z}\\] where $z$ is a complex number with $|z|=4$ .",
"role": "user"
}
] | 540 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Eight circles of radius $34$ can be placed tangent to $\\overline{BC}$ of $\\triangle ABC$ so that the circles are sequentially tangent to each other, with the first circle being ... | 197 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCD$ be a rhombus whose vertices all lie on the hyperbola $\\tfrac{x^2}{20}-\\tfrac{y^2}{24}=1$ and are in that order. If its diagonals intersect at the origin, find the lar... | 480 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 $P$ .... | 113 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 v... | 371 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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 ... | 385 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "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}+... | 110 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCD$ be a tetrahedron such that $AB = CD = \\sqrt{41}$ , $AC = BD = \\sqrt{80}$ , and $BC = AD = \\sqrt{89}$ . There exists a point $I$ inside the tetrahedron such that the ... | 104 |
[
{
"content": "Let's think step by step and output the final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$ . Let $r$ be the radius of the smallest sphere that can contain each of the rectangular b... | 721 |
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