prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider a graph $G$ with 2019 nodes. Define \"inverting\" a vertex $v$ as follows: for every other vertex $u$, if there was a... | 1018081 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f(x)=\\sin x\\cdot\\sin3x，若关于 x 的方程 f(x)=a 在 (0,\\pi] 上有奇数个不同的实数解，则实数 a 的值为__________.\n让我们一步一步地思考。请以“Answer: \\boxed{<fin... | -1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 f(x)=\\ln(1+x)(x>0)，定义函数序列 \\{f_n(x)\\}，其中 f_n 是 f 的 n 次迭代. 求所有的 x，使得 \\displaystyle\\lim_{n\\rightarrow+\\infty}nf_n(x) 存在... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the sum of all integers $n$ satisfying the following inequality:\n\n\\[\n\\frac{1}{4}<\\sin\\frac{\\pi}{n}<\\frac{1}{3}.\... | 33 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$f(x)=\\left\\{\\begin{array}{ll}x^{a} \\sin \\left(\\frac{1}{x^{3}}\\right) & x>0, a \\in N_{+} \\\\ 0 & x \\leq 0\\end{array... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six ... | 11 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numb... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. ... | 420 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor what integers $n \\geq 3$ is it possible to arrange the numbers $1, 2, \\ldots, n$ in a circular order such that every num... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a circle with radius $10$, point $M$ is located on the chord $PQ$ such that $PM = 5$ and $MQ = 10$. Through point $M$, we d... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe nonzero real numbers $a, b, c$ satisfy the following system of equations:\n\\[\n\\begin{cases} \n a + ab = c \\\\\n ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursively as $... | 256 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the minimum possible amount of distinct prime divisors of $19^{4n} + 4$, where $n$ is a positive integer.\n\nRemembe... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p, q$ be prime numbers such that $n^{3pq} - n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possi... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each with at ... | 30 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any natural number, let $S(n)$ denote the sum of the digits of $n$. Find the number of 3-digit numbers for which $S(S(n)) ... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the largest $2$ -digit prime factor of the integer $n = {200\\choose 100}$ ?\n\nRemember to put your answer on its own... | 61 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the cir... | 175 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the value of $a_2+a_4+a_6+a_8+\\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\\ldots$ is an arithmetic progression with common diff... | 93 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic me... | 649 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe equation $z^6+z^3+1=0$ has complex roots with argument $\\theta$ between $90^\\circ$ and $180^\\circ$ in the complex plane... | 160 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x$ . If $x=0$ is a r... | 401 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine $x^2+y^2+z^2+w^2$ if \n$\\frac{x^2}{2^2-1}+\\frac{y^2}{2^2-3^2}+\\frac{z^2}{2^2-5^2}+\\frac{w^2}{2^2-7^2}=1$ \n$\\fr... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$. What is t... | 986 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sum of the following seven numbers is exactly $19$: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.7... | 61 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its ma... | 85 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting fr... | 182 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe numbers in the sequence $101$, $104$, $109$, $116$,$\\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\\ldots$ For ea... | 401 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate the product \\[\\left(\\sqrt{5}+\\sqrt{6}+\\sqrt{7}\\right)\\left(\\sqrt{5}+\\sqrt{6}-\\sqrt{7}\\right)\\left(\\sqrt{... | 104 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the sum of the base $10$ logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000... | 141 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P... | 306 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$ , $b$ , and $c$... | 358 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$ . Given that the length of the hypotenuse $AB... | 400 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nBy a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural numb... | 182 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute\n\\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]\... | 373 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $(\\log_2 x)^2$ if $\\log_2 (\\log_8 x) = \\log_8 (\\log_2 x)$ .\n\nRemember to put your answer on its own line after \"A... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $w_1, w_2, \\dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_... | 163 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute $\\sqrt{(31)(30)(29)(28)+1}$.\n\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"... | 869 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be a subset of $\\{1,2,3,\\cdots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest n... | 905 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $n$ , it can be shown that every complex number of the form $r+si$ , where $r$ and $s$ are integers, ... | 490 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the value of $(52+6\\sqrt{43})^{3/2}-(52-6\\sqrt{43})^{3/2}$ .\n\nRemember to put your answer on its own line after \"Ans... | 828 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA triangle has vertices $P_{}^{}=(-8,5)$ , $Q=(-15,-19)$ , and $R=(1,-7)$ . The equation of the bisector of $\\angle P$ can be... | 89 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two target... | 560 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sets $A = \\{z : z^{18} = 1\\}$ and $B = \\{w : w^{48} = 1\\}$ are both sets of complex roots of unity . The set $C = \\{z... | 144 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can b... | 720 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $T = \\{9^k : k ~ {is ~an~ integer}, 0 \\le k \\le 4000\\}$ . Given that $9^{4000}$ has 3817 digits and that its first (le... | 184 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \\sqrt{3}$ and $BC^{}_{} = 13 \\sqrt{3}$. Diagonals $\\overlin... | 594 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind $A^2$ , where $A$ is the sum of the absolute values of all roots of the following equation:\n$x = \\sqrt{19} + \\frac{91}... | 383 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $\\sec x+\\tan x=\\frac{22}7$ and that $\\csc x+\\cot x=\\frac mn,$ where $\\frac mn$ is in lowest terms. Find $m... | 44 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nRhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$ , $Q$ , $R$ , and $S$ are interior points on sides $\\ove... | 677 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selecte... | 990 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA hexagon is inscribed in a circle . Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$ , has lengt... | 384 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor positive integer $n_{}^{}$, define $S_n^{}$ to be the minimum value of the sum\n$\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},$\nw... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTrapezoid $ABCD$ has sides $AB=92$ , $BC=50$ , $CD=19$ , and $AD=70$ , with $AB$ parallel to $CD$ . A circle with center $P$ o... | 164 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representatio... | 396 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are label... | 344 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R\\,$ and $S\\,$ be the points where the circles inscribed in th... | 997 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle with diameter $\\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is c... | 312 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the positive integer $n$ for which $$\\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\c... | 312 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a positive integer $n\\,$, let $p(n)\\,$ be the product of the non-zero digits of $n\\,$. (If $n\\,$ has only one digit... | 103 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\\overline{AB}$ at $D.$ The lengths of the side... | 450 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects wit... | 71 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,... | 597 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the last three digits of the product of the positive roots of $\\sqrt{1995}x^{\\log_{1995}x}=x^2$ .\n\nRemember to put yo... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$. The c... | 224 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $(1+\\sin t)(1+\\cos t)=5/4$ and\n\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\nwhere $k, m,$ and $n_{}$ are pos... | 27 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.... | 40 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads befo... | 37 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a magic square , the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of t... | 200 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each real number $x$ , let $\\lfloor x \\rfloor$ denote the greatest integer that does not exceed $x$. For how many positi... | 340 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that... | 166 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+... | 23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer solution to $\\tan{19x^{\\circ}}=\\dfrac{\\cos{96^{\\circ}}+\\sin{96^{\\circ}}}{\\cos{96^{\... | 159 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply pl... | 126 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCircles of radii $5, 5, 8,$ and $\\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive i... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoint $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\\cdots A_n$, and $A_1A_2B$ is an equilateral triangle. W... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radi... | 198 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\nLet $S$ be the set of points in the Cartesian plane that satisfy \n$\\Big|\\big||x|-2\\big|-1\\Big|+\\Big|\\big||y|-2\\big|-... | 66 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$ . Let $\\frac{m}{n}$ be the probability that... | 582 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region?\n\n\... | 800 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$\n\nRemember to ... | 40 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between ... | 87 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\\overline{BC}, \\overline{CA},$ and $\\overline{AB},$ resp... | 83 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ ... | 314 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let u... | 25 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTen points in the plane are given, with no three collinear . Four distinct segments joining pairs of these points are chosen a... | 489 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPoint $P_{}$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triang... | 463 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers,... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number\n\n$\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}}$\ncan be written as $\\frac mn$ where $m$ and $n$ are rel... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2... | 98 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ a... | 667 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble ... | 26 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe system of equations \n$$\\begin{aligned}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\... | 25 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$, find the least integer that is greater than $z^{20... | 0 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of numbers $x_{1},x_{2},x_{3},\\cdots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ ... | 173 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle is inscribed in quadrilateral $ABCD$, tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$. Given that $... | 647 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What ... | 248 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a function $f$ for which \n$f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$ \nholds for all real $x,$ what is the large... | 177 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvery positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\\ldots,f_m)$, meaning that $k=1!\\cdot f_1+2!\... | 495 |
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