prompt list | label string |
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[
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"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nFind the largest possible real part of \\[(75+117i)z+\\frac{96+144i}{z}\\]where $z$ is a complex number with $|z|=4$.\n\nRemember to put your final answer within \\boxed{}.",
"role": "user"
}
] | 540 |
[
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"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nEvery 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 $t$ minu... | 204 |
[
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"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $\\mathcal{B}$ 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 boxes that are elements ... | 721 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nA list of positive integers has the following properties:\n$\\bullet$ The sum of the items in the list is $30$.\n$\\bullet$ The unique mode of the list is $9$.\n$\\bullet$ The median of the list is a p... | 236 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nAlice 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 either $1$ token or $4$ tokens from the st... | 809 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nFind 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 formed by reading top t... | 45 |
[
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"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nDefine $f(x)=|| x|-\\tfrac{1}{2}|$ 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))).... | 385 |
[
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"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nRectangles $ABCD$ 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 $CE$?\n\nRemember to put ... | 104 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nFind the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}\n\nReme... | 601 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet \\(O=(0,0)\\), \\(A=\\left(\\tfrac{1}{2},0\\right)\\), and \\(B=\\left(0,\\tfrac{\\sqrt{3}}{2}\\right)\\) be points in the coordinate plane. Let \\(\\mathcal{F}\\) be the family of segments \\(\\ov... | 23 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when\n\\[\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\]\nis divided by 1000.\n\nRemember to put your final answer within \\boxed{}.",
... | 321 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $ABCD$ 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 distances from $I$ to each of the ... | 104 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nThere is a collection of $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in the $25$ unit cells of a $5\\times5$ grid suc... | 902 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet \\(b\\ge 2\\) be an integer. Call a positive integer \\(n\\) \\(b\\text-\\textit{eautiful}\\) if it has exactly two digits when expressed in base \\(b\\) and these two digits sum to \\(\\sqrt n\\)... | 211 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $ABC$ 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 $... | 113 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}... | 33 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $p$ 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^{... | 110 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nThere exist real numbers $x$ and $y$, both greater than 1, such that $\\log_x\\left(y^x\\right)=\\log_y\\left(x^{4y}\\right)=10$. Find $xy$.\n\nRemember to put your final answer within \\boxed{}.",
... | 25 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nAmong 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 residents owns a bag of candy h... | 73 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nAlice 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 $A$. Bob's list has 2024 sets... | 55 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nTorus $\\mathcal T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane a distance $6$ from the center of the circle. When a sphere of radius $11$ rests inside $\\... | 127 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nConsider 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 direction exactly four ti... | 294 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.\n\nRemember to put your final answer with... | 468 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nEight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same mann... | 197 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $A$, $B$, $C$, and $D$ be point on the hyperbola $\\frac{x^2}{20}- \\frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is ... | 480 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet $N$ be the greatest four-digit positive 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 quotient and re... | 699 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nFind 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 dodecagon. The diagram below sho... | 315 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nEach 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 posi... | 371 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nLet ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find... | 80 |
[
{
"content": "Solve the following math problem step by step. Please put your final answer within \\boxed{}.\n\nJen 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 were $2$ of the randoml... | 116 |
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