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\nCalculate the number of ways to choose $4$ numbers from the set $\\{1, 2, \\ldots, 11\\}$ such that at least $2$ of the number... | 260 |
[
{
"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 the set $S = \\{ xy(x + y) \\mid x, y \\in \\mathbb{N} \\}$. Let $a$ and $n$ be natural numbers such that $a + 2^k \\in ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlbert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous n... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSome vertices of a regular $2024$-gon are marked such that for any regular polygon, all of whose vertices are vertices of the ... | 506 |
[
{
"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 distinct positive integers $a$ and $b$ have the property that $\\frac{a+b}{2}$, $\\sqrt{ab}$, and $\\frac{2}{\\frac{1}{a}+... | 30 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $f$ be a non-constant polynomial such that \n\\[ f(x-1) + f(x) + f(x+1) = \\frac{f(x)^2}{2013x} \\] \nfor all nonzero real... | 6039 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n25 little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle $C_0$ is inscribed in an equilateral triangle $XYZ$ with a side length of 112. For each positive integer $n$, circle ... | 1176 |
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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\nThere are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-tu... | 1009 |
[
{
"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 positive integers $1 \\leq k \\leq 99$ such that there exist positive integers $a$ and $b$ with the proper... | 117 |
[
{
"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 $n$ which cannot be expressed as \\( n = \\frac{2^a - 2^b}{2^c - 2^d} \\) for some positive... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nPeter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting w... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn a group of $2020$ people, some pairs of people are friends (friendship is mutual). It is known that no two people (not nece... | 1010 |
[
{
"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 the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.\n\nRemember to put your answer on its own line after \"Answ... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet's call any natural number \"very prime\" if any number of consecutive digits (in particular, a digit or the number itself)... | 373 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to... | 22 |
[
{
"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 $p < q < r < s$ are prime numbers such that $pqrs + 1 = 4^{p+q}$. Find $r + s$.\n\nRemember to put your answer on its ... | 274 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the number of permutations of \\(\\{1,2,3\\}\\) with the property that there is some number that can be removed such t... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be a triangle with $BC = 20$ and $CA = 16$, and let $I$ be its incenter. If the altitude from $A$ to $BC$, the perpe... | 460 |
[
{
"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 greatest natural number $n$ such that there exist natural numbers $x_{1}, x_{2}, \\ldots, x_{n}$ and natural numbers ... | 6 |
[
{
"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 the number of ordered triples of integers $(a, b, c)$ between $1$ and $12$, inclusive, such that, if \\[ q = a + \\fra... | 132 |
[
{
"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 $(p_1, p_2, \\dots) = (2, 3, \\dots)$ be the list of all prime numbers, and $(c_1, c_2, \\dots) = (4, 6, \\dots)$ be the l... | 16 |
[
{
"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 real solutions to $(x^2 - 10x - 12)^{x^2+5x+2} = 1$.\n\nRemember to put your answer on its own line after ... | 15 |
[
{
"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 following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \\dots, 2^{100} - 1.$$ \n\nAlice chooses t... | 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\nBy default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal... | 21600 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nI have 6 friends and during a vacation, I met them during several dinners. I found that:\n\n- I dined with all 6 friends exact... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe product $20! \\cdot 21! \\cdot 22! \\cdot \\ldots \\cdot 28!$ can be expressed in the form $m \\cdot n^3$, where $m$ and $... | 825 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many ways are there to color the vertices of a triangle using the colors red, green, blue, or yellow such that no two vert... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCompute the area of the trapezoid $ABCD$ with right angles $\\angle BAD$ and $\\angle ADC$. The side lengths are given as $AB ... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\\frac{n}{k}$ is not an integer.... | 11 |
[
{
"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 $a_0 = 1$, $a_1 = 2$, and $a_n = 4a_{n-1} - a_{n-2}$ for $n \\ge 2$. Find an odd prime factor of $a_{2015}$.\n\nRemember t... | 181 |
[
{
"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\nMattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distribute... | 39 |
[
{
"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 $a$ such that:\n\n\\[101a = 6539\\int_{-1}^1 \\frac{x^{12}+31}{1+2011^{x}}\\ dx.\\]\n\nRemember to put your ... | 2012 |
[
{
"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(0) = 3$ and $F(n) = F(n-1) + 4$ when $n$ is positive. Find $F(F(F(5)))$.\n\nRemember to put your answer on its own line aft... | 383 |
[
{
"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 fraction $\\frac{1}{2015}$ has a unique \"(restricted) partial fraction decomposition\" of the form \\[\\frac{1}{2015} = \... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles have radii of 5 and 26, respectively. The smaller circle passes through the center of the larger one. What is the ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radiu... | 11 |
[
{
"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 cyclic quadrilateral with side lengths $1$, $4$, $8$, and $7$ in that order. What is its circumdiameter? Let the an... | 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 $f(x) = x^3 + ax + b$, where $a \\neq b$. Suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. F... | 1 |
[
{
"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 natural number $n$, you are allowed two operations: \n1. Multiply $n$ by $2$, or \n2. Subtract $3$ from $n$. \n\nFor exam... | 10 |
[
{
"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 $f(x) = x^2 + x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z... | 1735 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let th... | 457 |
[
{
"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 remainder when $2020!$ is divided by $2020^2$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 0 |
[
{
"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\nJon wrote the $n$ smallest perfect squares on one sheet of paper, and the $n$ smallest triangular numbers on another sheet (no... | 89 |
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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 quadruple $(a, b, c, d)$ of distinct integers is said to be $\\text{balanced}$ if $a + c = b + d$. Let $\\mathcal{S}$ be any... | 91 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest prime number that does not divide \\(9 + 9^2 + 9^3 + \\cdots + 9^{2010}\\).\n\nRemember to put your answer o... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest prime number that does not divide any five-digit number whose digits are in strictly increasing order.\... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo rays emanate from the origin $O$ and form a $45^\\circ$ angle in the first quadrant of the Cartesian coordinate plane. For... | 1500 |
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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 $Answer is the answer to the problem.\n\nEvaluate \\( \\int_0^{2008} \\left(3x^2 - 8028x + 2007^2 + \\frac{1}{2008}\\right)\\, dx \\).\n\nRemember to put your answer o... | 2009 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe number $21982145917308330487013369$ is the thirteenth power of a positive integer. Find the positive integer.\n\nRemember ... | 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\nA crazy physicist has discovered a new particle called an omon. He has a machine that takes two omons of mass $a$ and $b$ and ... | 9 |
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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 $Answer is the answer to the problem.\n\nObserving the temperatures recorded in Cesenatico during December and January, Stefano noticed an interesting coincidence: for... | -3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \\geq 1$, ... | 5 |
[
{
"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 a planet contains $(CCAMATHBONANZA_{71})^{100}$ people ($100$ in decimal), where in base $71$ the digits $A, B, C... | 8355 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA positive integer with at least two digits written in base $10$ is called 'ascending' if its digits increase in value from le... | 502 |
[
{
"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 number of integers $n$ for which $n^2 + 10n < 2008$.\n\nRemember to put your answer on its own line after \"Answer:\"... | 91 |
[
{
"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\nOne side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is t... | 1875 |
[
{
"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 semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the dia... | 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\nThe polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0$. What is $a + b + c... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of partitions of the set $\\{1,2,3,\\cdots ,11,12\\}$ into three nonempty subsets such that no subset has two ... | 1023 |
[
{
"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\nDaniel chooses some distinct subsets of $\\{1, \\dots, 2019\\}$ such that any two distinct subsets chosen are disjoint. Comput... | 2020 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe sequence \\( \\{c_{n}\\} \\) is determined by the following equation: \n\\[ c_{n} = (n+1) \\int_{0}^{1} x^{n} \\cos \\pi ... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.\n\nRemem... | 43 |
[
{
"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\nTriangle $\\triangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $BC$, and ... | 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\nTanya wrote numbers in the form $n^7 - 1$ for $n = 2, 3, \\ldots$ and noticed that for $n = 8$, she obtained a number divisibl... | 79 |
[
{
"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 Flyfishing Club is choosing officers. There are $23$ members in the club: $14$ boys and $9$ girls. In how many ways can th... | 252 |
[
{
"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 duck language, only the letters $q$, $a$, and $k$ are used. There is no word with two consonants in succession, as ducks ca... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nYou are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open ... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\triangle ABC$ be a triangle where $\\angle ACB = 40^\\circ$ and $\\angle BAC = 60^\\circ$. Let $D$ be a point inside th... | 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\nFind the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward... | 504405 |
[
{
"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 such that when the last digit is moved to the front, the new number is $\\frac{3}{2}$ times... | 285714 |
[
{
"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\nOne day, I went for a walk in the morning at $x$ minutes past 5 o'clock, where $x$ is a two-digit number. When I returned, it ... | 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\nHow many integers can be expressed in the form: $\\pm 1 \\pm 2 \\pm 3 \\pm 4 \\pm \\cdots \\pm 2018$?\n\nRemember to put your ... | 2037172 |
[
{
"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 $2 \\times 8$ squared board, you want to color each square red or blue such that on each $2 \\times 2$ sub-board there ar... | 341 |
[
{
"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 the sequence of positive integers $\\{a_n\\}$ as follows:\n- Let $a_1=1$, $a_2=3$.\n- For each $n>2$, let $a_n$ be the ... | 23097 |
[
{
"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\nIf $a_1, a_2, \\ldots$ is a sequence of real numbers such that for all $n$, $$\\sum_{k = 1}^n a_k \\left( \\frac{k}{n} \\righ... | 4036 |
[
{
"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 concentric circles $\\omega$ and $\\Omega$ have radii 8 and 13, respectively. $AB$ is a diameter of $\\Omega$, and the tan... | 19 |
[
{
"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 $2018 \\times 4$ grid, tint the cells with red and blue such that each row and each column has an equal number of red ... | 6 |
[
{
"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\nHow many ways are there to remove an $11 \\times 11$ square from a $2011 \\times 2011$ square so that the remaining part can b... | 2002001 |
[
{
"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 natural number $n$, define the function $f(n)$ as the sum of the digits of the number $n$. For example, $f(16) = 7$, $f(... | 20000000000000000000000 |
[
{
"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 $f(x) = x^4 + |x|$, let $I_1 = \\int_0^\\pi f(\\cos x)\\, dx$ and $I_2 = \\int_0^{\\frac{\\pi}{2}} f(\\sin x)\\, dx$. Find... | 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\nHow many positive integers $n$ are there such that $3 \\leq n \\leq 100$ and $x^{2^n} + x + 1$ is divisible by $x^2 + x + 1$?\... | 49 |
[
{
"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 number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \\ge 0.$\n\nRemember to p... | 0 |
[
{
"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 sequence $\\{a_n\\}_{n\\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$... | 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\nA paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the t... | 93 |
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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 last two digits of the number $2^5 + 2^{5^2} + 2^{5^3} + \\ldots + 2^{5^{1991}}$, written in decimal notation.\n... | 12 |
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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 $2020 \\times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square ... | 2020 |
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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\nHow many positive integers $n \\leq 100$ satisfy $\\lfloor n\\pi \\rfloor = \\lfloor (n-1)\\pi \\rfloor + 3$? Here $\\lfloor x... | 86 |
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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 $a, b, c$ are complex numbers satisfying the system of equations: \n\\[a^2 = b - c,\\] \n\\[b^2 = c - a,\\] \n... | 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\nThe first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s,... | 13 |
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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 minimum positive integer $n$ such that for any set $A$ with $n$ positive integers, there exists a subset of 15 elemen... | 29 |
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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\nHow many different solutions are there on the interval $[0, \\pi]$ for the equation: $$6\\sqrt{2} \\sin x \\cdot \\tan x - 2\\... | 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\nLet $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is... | 28 |
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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\nFive identical circles are placed in a line inside a larger circle as shown. If the shown chord has a length of $16$, find the... | 10 |
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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 greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and... | 2093124 |
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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 minimum of $x + y + z$ where $x$, $y$, and $z$ are real numbers satisfying the following conditions:\n- $x \\geq 4$\n... | 16 |
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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\nAndy has 2010 square tiles, each with a side length of one unit. He plans to arrange the tiles in an $m \\times n$ rectangle, ... | 9792 |
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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 triangle $ABC$, where $AB = 13$, $BC = 14$, and $CA = 15$, the segment $BC$ is divided into $n+1$ congruent segments by $n$... | 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 $x_1$, $x_2$, \\ldots, $x_{10}$ be 10 numbers. Suppose that $x_i + 2x_{i+1} = 1$ for each $i$ from 1 through 9. What is th... | 171 |
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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\nHow many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of thi... | 114 |
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