AIMO
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Conquering worlds' hardest math challenges with LLMs.
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When the roots of the polynomial
\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]
are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?
| 6 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhen the roots of the polynomial\n\\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}\\]\nare removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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] |
In equiangular octagon $CAROLINE$ $CA = RO = LI = NE =$ $\sqrt{2}$ and $AR = OL = IN = EC = 1$ . The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$ , that is, the total area of the six triangular regions. Then $K =$ $\dfrac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$ | 23 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn equiangular octagon $CAROLINE$ $CA = RO = LI = NE =$ $\\sqrt{2}$ and $AR = OL = IN = EC = 1$ . The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$ , that is, the total area of the six triangular regions. Then $K =$ $\\dfrac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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In convex quadrilateral $KLMN$ , side $\overline{MN}$ is perpendicular to diagonal $\overline{KM}$ , side $\overline{KL}$ is perpendicular to diagonal $\overline{LN}$ , $MN = 65$ , and $KL = 28$ . The line through $L$ perpendicular to side $\overline{KN}$ intersects diagonal $\overline{KM}$ at $O$ with $KO = 8$ . Find $MO$ . | 90 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn convex quadrilateral $KLMN$ , side $\\overline{MN}$ is perpendicular to diagonal $\\overline{KM}$ , side $\\overline{KL}$ is perpendicular to diagonal $\\overline{LN}$ , $MN = 65$ , and $KL = 28$ . The line through $L$ perpendicular to side $\\overline{KN}$ intersects diagonal $\\overline{KM}$ at $O$ with $KO = 8$ . Find $MO$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$ , where $a$ and $b$ are positive integers. What is $a+b$
| 2,023 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe sum \\[\\frac{1}{2!}+\\frac{2}{3!}+\\frac{3}{4!}+\\cdots+\\frac{2021}{2022!}\\] can be expressed as $a-\\frac{1}{b!}$ , where $a$ and $b$ are positive integers. What is $a+b$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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In triangle $ABC$ , $AB=\sqrt{30}$ , $AC=\sqrt{6}$ , and $BC=\sqrt{15}$ . There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ , and $\angle ADB$ is a right angle. The ratio \[\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}\] can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 65 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn triangle $ABC$ , $AB=\\sqrt{30}$ , $AC=\\sqrt{6}$ , and $BC=\\sqrt{15}$ . There is a point $D$ for which $\\overline{AD}$ bisects $\\overline{BC}$ , and $\\angle ADB$ is a right angle. The ratio \\[\\dfrac{\\text{Area}(\\triangle ADB)}{\\text{Area}(\\triangle ABC)}\\] can be written in the form $\\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Someone observed that $6! = 8 \cdot 9 \cdot 10$ . Find the largest positive integer $n^{}_{}$ for which $n^{}_{}!$ can be expressed as the product of $n - 3_{}^{}$ consecutive positive integers. | 23 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nSomeone observed that $6! = 8 \\cdot 9 \\cdot 10$ . Find the largest positive integer $n^{}_{}$ for which $n^{}_{}!$ can be expressed as the product of $n - 3_{}^{}$ consecutive positive integers.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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A geometric sequence $(a_n)$ has $a_1=\sin x$ $a_2=\cos x$ , and $a_3= \tan x$ for some real number $x$ . For what value of $n$ does $a_n=1+\cos x$
| 8 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA geometric sequence $(a_n)$ has $a_1=\\sin x$ $a_2=\\cos x$ , and $a_3= \\tan x$ for some real number $x$ . For what value of $n$ does $a_n=1+\\cos x$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$ . How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$
| 1,011 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$ . How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The point in the $xy$ -plane with coordinates $(1000, 2012)$ is reflected across the line $y=2000$ . What are the coordinates of the reflected point?
| 10,001,988 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe point in the $xy$ -plane with coordinates $(1000, 2012)$ is reflected across the line $y=2000$ . What are the coordinates of the reflected point?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?
$\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000$ | 400,000 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?\n$\\mathrm{(A)}\\ 270,000\\qquad\\mathrm{(B)}\\ 385,000\\qquad\\mathrm{(C)}\\ 400,000\\qquad\\mathrm{(D)}\\ 430,000\\qquad\\mathrm{(E)}\\ 700,000$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The fraction
\[\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}\] simplifies to which of the following?
$\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ \frac{9}{4}\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ \frac{9}{2}\qquad\mathrm{(E)}\ 9$ | 9 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe fraction\n\\[\\frac{\\left(3^{2008}\\right)^2-\\left(3^{2006}\\right)^2}{\\left(3^{2007}\\right)^2-\\left(3^{2005}\\right)^2}\\] simplifies to which of the following?\n$\\mathrm{(A)}\\ 1\\qquad\\mathrm{(B)}\\ \\frac{9}{4}\\qquad\\mathrm{(C)}\\ 3\\qquad\\mathrm{(D)}\\ \\frac{9}{2}\\qquad\\mathrm{(E)}\\ 9$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\leq$ $k$ $\leq$ $5$ , at least one of the first $k$ terms of the permutation is greater than $k$ | 461 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\\leq$ $k$ $\\leq$ $5$ , at least one of the first $k$ terms of the permutation is greater than $k$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \ldots, 30$ such that for $m \in \left\{{2, 3, 5}\right\}$ $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \leq n < n+m \leq 30$ . Find the remainder when $N$ is divided by $1000$ | 440 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThere are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$ $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$ . Find the remainder when $N$ is divided by $1000$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom?
$\mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15$ | 10 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nRiders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom?\n$\\mathrm{(A) \\ } 5\\qquad \\mathrm{(B) \\ } 6\\qquad \\mathrm{(C) \\ } 7.5\\qquad \\mathrm{(D) \\ } 10\\qquad \\mathrm{(E) \\ } 15$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$ , and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\]
Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$ . What is the sum of all integers $k$ $1\le k \le 2011$ , such that $a_k=b_k?$
| 1,341 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $\\{a_k\\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$ , and in general,\n\\[a_k=\\begin{cases}(0.\\underbrace{20101\\cdots 0101}_{k+2\\text{ digits}})^{a_{k-1}}\\qquad\\text{if }k\\text{ is odd,}\\\\(0.\\underbrace{20101\\cdots 01011}_{k+2\\text{ digits}})^{a_{k-1}}\\qquad\\text{if }k\\text{ is even.}\\end{cases}\\]\nRearranging the numbers in the sequence $\\{a_k\\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\\{b_k\\}_{k=1}^{2011}$ . What is the sum of all integers $k$ $1\\le k \\le 2011$ , such that $a_k=b_k?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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For every integer $n\ge2$ , let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$ . For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$ . What is the largest integer $m$ such that $2010^m$ divides
| 77 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor every integer $n\\ge2$ , let $\\text{pow}(n)$ be the largest power of the largest prime that divides $n$ . For example $\\text{pow}(144)=\\text{pow}(2^4\\cdot3^2)=3^2$ . What is the largest integer $m$ such that $2010^m$ divides\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$ . What is the area of $S_3$
$\mathrm{(A)}\ \frac{1}{2}\qquad\mathrm{(B)}\ 1\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ 4$ | 4 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nEach of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$ . What is the area of $S_3$\n$\\mathrm{(A)}\\ \\frac{1}{2}\\qquad\\mathrm{(B)}\\ 1\\qquad\\mathrm{(C)}\\ 2\\qquad\\mathrm{(D)}\\ 3\\qquad\\mathrm{(E)}\\ 4$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Circle $C$ with radius 2 has diameter $\overline{AB}$ . Circle $D$ is internally tangent to circle $C$ at $A$ . Circle $E$ is internally tangent to circle $C$ , externally tangent to circle $D$ , and tangent to $\overline{AB}$ . The radius of circle $D$ is three times the radius of circle $E$ , and can be written in the form $\sqrt{m}-n$ , where $m$ and $n$ are positive integers. Find $m+n$ . | 254 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nCircle $C$ with radius 2 has diameter $\\overline{AB}$ . Circle $D$ is internally tangent to circle $C$ at $A$ . Circle $E$ is internally tangent to circle $C$ , externally tangent to circle $D$ , and tangent to $\\overline{AB}$ . The radius of circle $D$ is three times the radius of circle $E$ , and can be written in the form $\\sqrt{m}-n$ , where $m$ and $n$ are positive integers. Find $m+n$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$ . Including $\overline{AB}$ and $\overline{BC}$ , how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$
| 13 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nRight triangle $ABC$ has leg lengths $AB=20$ and $BC=21$ . Including $\\overline{AB}$ and $\\overline{BC}$ , how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\\overline{AC}$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Let $\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$ | 540 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $\\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Let $f_n (x) = \text{sin}^n x + \text{cos}^n x.$ For how many $x$ in $[0,\pi]$ is it true that
\[6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?\]
| 8 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $f_n (x) = \\text{sin}^n x + \\text{cos}^n x.$ For how many $x$ in $[0,\\pi]$ is it true that\n\\[6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?\\]\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]
| 2 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nDefine $x\\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \\[(1\\diamond(2\\diamond3))-((1\\diamond2)\\diamond3)?\\]\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$ , $0^{}_{}<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$ , where the digits $a^{}_{}$ , $b^{}_{}$ , and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{}$ as fractions in lowest terms, how many different numerators are required? | 660 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $S^{}_{}$ be the set of all rational numbers $r^{}_{}$ , $0^{}_{}<r<1$ , that have a repeating decimal expansion in the form $0.abcabcabc\\ldots=0.\\overline{abc}$ , where the digits $a^{}_{}$ , $b^{}_{}$ , and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{}$ as fractions in lowest terms, how many different numerators are required?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\cdot N$ contains no square of an integer. | 282 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\\cdot N$ contains no square of an integer.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$ . He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens? | 342 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$ . He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The parallelogram bounded by the lines $y=ax+c$ $y=ax+d$ $y=bx+c$ , and $y=bx+d$ has area $18$ . The parallelogram bounded by the lines $y=ax+c$ $y=ax-d$ $y=bx+c$ , and $y=bx-d$ has area $72$ . Given that $a$ $b$ $c$ , and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$
$\mathrm {(A)} 13\qquad \mathrm {(B)} 14\qquad \mathrm {(C)} 15\qquad \mathrm {(D)} 16\qquad \mathrm {(E)} 17$ | 16 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe parallelogram bounded by the lines $y=ax+c$ $y=ax+d$ $y=bx+c$ , and $y=bx+d$ has area $18$ . The parallelogram bounded by the lines $y=ax+c$ $y=ax-d$ $y=bx+c$ , and $y=bx-d$ has area $72$ . Given that $a$ $b$ $c$ , and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$\n$\\mathrm {(A)} 13\\qquad \\mathrm {(B)} 14\\qquad \\mathrm {(C)} 15\\qquad \\mathrm {(D)} 16\\qquad \\mathrm {(E)} 17$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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] |
Let
\[p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.\]
Suppose that
\[p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)\\ = p(1,1) = p(1, - 1) = p(2,2) = 0.\]
There is a point $\left(\frac {a}{c},\frac {b}{c}\right)$ for which $p\left(\frac {a}{c},\frac {b}{c}\right) = 0$ for all such polynomials, where $a$ $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . Find $a + b + c$ | 40 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet\n\\[p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.\\]\nSuppose that\n\\[p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)\\\\ = p(1,1) = p(1, - 1) = p(2,2) = 0.\\]\nThere is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\frac {a}{c},\\frac {b}{c}\\right) = 0$ for all such polynomials, where $a$ $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . Find $a + b + c$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
$\mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45$ | 18 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?\n$\\mathrm{(A) \\ } 9 \\qquad \\mathrm{(B) \\ } 18 \\qquad \\mathrm{(C) \\ } 27 \\qquad \\mathrm{(D) \\ } 36\\qquad \\mathrm{(E) \\ } 45$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Triangle $ABC$ with right angle at $C$ , $\angle BAC < 45^\circ$ and $AB = 4$ . Point $P$ on $\overline{AB}$ is chosen such that $\angle APC = 2\angle ACP$ and $CP = 1$ . The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$ , where $p$ , $q$ , $r$ are positive integers and $r$ is not divisible by the square of any prime. Find $p+q+r$ . | 7 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTriangle $ABC$ with right angle at $C$ , $\\angle BAC < 45^\\circ$ and $AB = 4$ . Point $P$ on $\\overline{AB}$ is chosen such that $\\angle APC = 2\\angle ACP$ and $CP = 1$ . The ratio $\\frac{AP}{BP}$ can be represented in the form $p + q\\sqrt{r}$ , where $p$ , $q$ , $r$ are positive integers and $r$ is not divisible by the square of any prime. Find $p+q+r$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
| 1,022 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA scanning code consists of a $7 \\times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?
| 32 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let $M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $M$ is divided by $10$ | 375 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nOn a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let $M$ be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when $M$ is divided by $10$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Triangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\ell_A,\ell_B$ , and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$ , and $\overline{AB}$ , respectively, such that the intersections of $\ell_A,\ell_B$ , and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\ell_A,\ell_B$ , and $\ell_C$ | 715 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTriangle $ABC$ has side lengths $AB=120,BC=220$ , and $AC=180$ . Lines $\\ell_A,\\ell_B$ , and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$ , and $\\overline{AB}$ , respectively, such that the intersections of $\\ell_A,\\ell_B$ , and $\\ell_C$ with the interior of $\\triangle ABC$ are segments of lengths $55,45$ , and $15$ , respectively. Find the perimeter of the triangle whose sides lie on lines $\\ell_A,\\ell_B$ , and $\\ell_C$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$ , and a different number in $S$ was divisible by $7$ . The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.
However, upon hearing that all four students replied no, each student was able to determine the elements of $S$ . Find the sum of all possible values of the greatest element of $S$ | 258 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$ , and a different number in $S$ was divisible by $7$ . The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.\nHowever, upon hearing that all four students replied no, each student was able to determine the elements of $S$ . Find the sum of all possible values of the greatest element of $S$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
| 25 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAt Typico High School, $60\\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
| 30 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nJerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$ | 50 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A circle with integer radius $r$ is centered at $(r, r)$ . Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$ $b_i$ , and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$ . What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$
| 17 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA circle with integer radius $r$ is centered at $(r, r)$ . Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \\le i \\le 14$ and are tangent to the circle, where $a_i$ $b_i$ , and $c_i$ are all positive integers and $c_1 \\le c_2 \\le \\cdots \\le c_{14}$ . What is the ratio $\\frac{c_{14}}{c_1}$ for the least possible value of $r$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer? | 215 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhat is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For how many pairs of consecutive integers in $\{1000,1001,1002,\ldots,2000\}$ is no carrying required when the two integers are added? | 156 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor how many pairs of consecutive integers in $\\{1000,1001,1002,\\ldots,2000\\}$ is no carrying required when the two integers are added?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?
| 0 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many of the first ten numbers of the sequence $121, 11211, 1112111, \\ldots$ are prime numbers?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$ | 510 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Positive integers $a$ $b$ $c$ , and $d$ satisfy $a > b > c > d$ $a + b + c + d = 2010$ , and $a^2 - b^2 + c^2 - d^2 = 2010$ . Find the number of possible values of $a$ | 501 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nPositive integers $a$ $b$ $c$ , and $d$ satisfy $a > b > c > d$ $a + b + c + d = 2010$ , and $a^2 - b^2 + c^2 - d^2 = 2010$ . Find the number of possible values of $a$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH HH , and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH , three HT , four TH , and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH , three HT , four TH , and five TT subsequences? | 560 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH HH , and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH , three HT , four TH , and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH , three HT , four TH , and five TT subsequences?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $f(n)$ and $g(n)$ be functions satisfying \[f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}\] and \[g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}\] for positive integers $n$ . Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$ | 258 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $f(n)$ and $g(n)$ be functions satisfying \\[f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 1 + f(n+1) & \\text{ otherwise} \\end{cases}\\] and \\[g(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 2 + g(n+2) & \\text{ otherwise} \\end{cases}\\] for positive integers $n$ . Find the least positive integer $n$ such that $\\tfrac{f(n)}{g(n)} = \\tfrac{4}{7}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$ . This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$ . Find the maximum possible area of $AQRS$ . | 744 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTriangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$ . This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$ . Find the maximum possible area of $AQRS$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For how many pairs of consecutive integers in $\{1000,1001,1002^{}_{},\ldots,2000\}$ is no carrying required when the two integers are added? | 156 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor how many pairs of consecutive integers in $\\{1000,1001,1002^{}_{},\\ldots,2000\\}$ is no carrying required when the two integers are added?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$ , and $\overline{AB}$ is parallel to $\overline{ED}$ . The angles $AEB$ and $ABE$ are in the ratio $4 : 5$ . What is the degree measure of angle $BCD$
| 130 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn the given circle, the diameter $\\overline{EB}$ is parallel to $\\overline{DC}$ , and $\\overline{AB}$ is parallel to $\\overline{ED}$ . The angles $AEB$ and $ABE$ are in the ratio $4 : 5$ . What is the degree measure of angle $BCD$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
An object moves $8$ cm in a straight line from $A$ to $B$ , turns at an angle $\alpha$ , measured in radians and chosen at random from the interval $(0,\pi)$ , and moves $5$ cm in a straight line to $C$ . What is the probability that $AC < 7$
$\mathrm{(A)}\ \frac{1}{6} \qquad\mathrm{(B)}\ \frac{1}{5} \qquad\mathrm{(C)}\ \frac{1}{4} \qquad\mathrm{(D)}\ \frac{1}{3} \qquad\mathrm{(E)}\ \frac{1}{2}$ | 13 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAn object moves $8$ cm in a straight line from $A$ to $B$ , turns at an angle $\\alpha$ , measured in radians and chosen at random from the interval $(0,\\pi)$ , and moves $5$ cm in a straight line to $C$ . What is the probability that $AC < 7$\n$\\mathrm{(A)}\\ \\frac{1}{6} \\qquad\\mathrm{(B)}\\ \\frac{1}{5} \\qquad\\mathrm{(C)}\\ \\frac{1}{4} \\qquad\\mathrm{(D)}\\ \\frac{1}{3} \\qquad\\mathrm{(E)}\\ \\frac{1}{2}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$ . Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit? | 184 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $T = \\{9^k : k ~ \\mbox{is an integer}, 0 \\le k \\le 4000\\}$ . Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In triangle $ABC$ , $AB = AC = 100$ , and $BC = 56$ . Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$ . Circle $Q$ is externally tangent to circle $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$ . No point of circle $Q$ lies outside of $\bigtriangleup\overline{ABC}$ . The radius of circle $Q$ can be expressed in the form $m - n\sqrt{k}$ ,where $m$ , $n$ , and $k$ are positive integers and $k$ is the product of distinct primes. Find $m +nk$ . | 254 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn triangle $ABC$ , $AB = AC = 100$ , and $BC = 56$ . Circle $P$ has radius $16$ and is tangent to $\\overline{AC}$ and $\\overline{BC}$ . Circle $Q$ is externally tangent to circle $P$ and is tangent to $\\overline{AB}$ and $\\overline{BC}$ . No point of circle $Q$ lies outside of $\\bigtriangleup\\overline{ABC}$ . The radius of circle $Q$ can be expressed in the form $m - n\\sqrt{k}$ ,where $m$ , $n$ , and $k$ are positive integers and $k$ is the product of distinct primes. Find $m +nk$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$ $BC = 14$ , and $AD = 2\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$ . Find the area of quadrilateral $ABCD$ | 112 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $ABCD$ be a convex quadrilateral with $AB = CD = 10$ $BC = 14$ , and $AD = 2\\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$ . Find the area of quadrilateral $ABCD$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
$\mathrm {(A)} 6\qquad \mathrm {(B)} 7\qquad \mathrm {(C)} 8\qquad \mathrm {(D)} 10\qquad \mathrm {(E)} 12$ | 6 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?\n$\\mathrm {(A)} 6\\qquad \\mathrm {(B)} 7\\qquad \\mathrm {(C)} 8\\qquad \\mathrm {(D)} 10\\qquad \\mathrm {(E)} 12$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
$\mathrm{(A) \ } 8 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 16 \qquad \mathrm{(D) \ } 18 \qquad \mathrm{(E) \ } 24$ | 18 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAt a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?\n$\\mathrm{(A) \\ } 8 \\qquad \\mathrm{(B) \\ } 12 \\qquad \\mathrm{(C) \\ } 16 \\qquad \\mathrm{(D) \\ } 18 \\qquad \\mathrm{(E) \\ } 24$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\text{ }B,\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$ , and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\tfrac{2}{3}$ $\sin\varphi_B=\tfrac{3}{5}$ , and $\sin\varphi_C=\tfrac{6}{7}$ . All three quadrilaterals have the same area $K$ , which can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | 59 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nDavid found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$ , and define $\\varphi_B$ and $\\varphi_C$ similarly. Suppose that $\\sin\\varphi_A=\\tfrac{2}{3}$ $\\sin\\varphi_B=\\tfrac{3}{5}$ , and $\\sin\\varphi_C=\\tfrac{6}{7}$ . All three quadrilaterals have the same area $K$ , which can be written in the form $\\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The point $(-3,2)$ is rotated $90^\circ$ clockwise around the origin to point $B$ . Point $B$ is then reflected over the line $x=y$ to point $C$ . What are the coordinates of $C$
$\mathrm{(A)}\ (-3,-2) \qquad \mathrm{(B)}\ (-2,-3) \qquad \mathrm{(C)}\ (2,-3) \qquad \mathrm{(D)}\ (2,3) \qquad \mathrm{(E)}\ (3,2)$ | 32 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe point $(-3,2)$ is rotated $90^\\circ$ clockwise around the origin to point $B$ . Point $B$ is then reflected over the line $x=y$ to point $C$ . What are the coordinates of $C$\n$\\mathrm{(A)}\\ (-3,-2) \\qquad \\mathrm{(B)}\\ (-2,-3) \\qquad \\mathrm{(C)}\\ (2,-3) \\qquad \\mathrm{(D)}\\ (2,3) \\qquad \\mathrm{(E)}\\ (3,2)$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many ordered pairs $(a, b)$ of positive integers satisfy the equation \[a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),\] where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$ , and $\text{lcm}(a,b)$ denotes their least common multiple?
| 2 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many ordered pairs $(a, b)$ of positive integers satisfy the equation \\[a\\cdot b + 63 = 20\\cdot \\text{lcm}(a, b) + 12\\cdot\\text{gcd}(a,b),\\] where $\\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$ , and $\\text{lcm}(a,b)$ denotes their least common multiple?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?
| 2,320 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $R$ be a unit square region and $n \\geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray partitional?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$ , $a_{n+1}a_{n-1}=a_{n}^{2}+2007$ . Find the greatest integer that does not exceed $\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}$ | 224 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$ , $a_{n+1}a_{n-1}=a_{n}^{2}+2007$ . Find the greatest integer that does not exceed $\\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $f_{1}(x)=\sqrt{1-x}$ , and for integers $n \geq 2$ , let $f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})$ . If $N$ is the largest value of $n$ for which the domain of $f_{n}$ is nonempty, the domain of $f_{N}$ is $[c]$ . What is $N+c$
| 226 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $f_{1}(x)=\\sqrt{1-x}$ , and for integers $n \\geq 2$ , let $f_{n}(x)=f_{n-1}(\\sqrt{n^2 - x})$ . If $N$ is the largest value of $n$ for which the domain of $f_{n}$ is nonempty, the domain of $f_{N}$ is $[c]$ . What is $N+c$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Eight congruent equilateral triangles , each of a different color, are used to construct a regular octahedron . How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
$\textbf {(A)}\ 210 \qquad \textbf {(B)}\ 560 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 1260 \qquad \textbf {(E)}\ 1680$ | 1,680 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nEight congruent equilateral triangles , each of a different color, are used to construct a regular octahedron . How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)\n$\\textbf {(A)}\\ 210 \\qquad \\textbf {(B)}\\ 560 \\qquad \\textbf {(C)}\\ 840 \\qquad \\textbf {(D)}\\ 1260 \\qquad \\textbf {(E)}\\ 1680$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $A_1A_2A_3\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$ . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | 19 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$ . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ | 100 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \\cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A right hexagonal prism has height $2$ . The bases are regular hexagons with side length $1$ . Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles). | 52 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA right hexagonal prism has height $2$ . The bases are regular hexagons with side length $1$ . Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For each positive integer $n,$ let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$ . Find the largest value of $n$ for which $f(n) \le 300$ . Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ . | 109 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor each positive integer $n,$ let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$ . Find the largest value of $n$ for which $f(n) \\le 300$ . Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For a positive integer $p$ , define the positive integer $n$ to be $p$ -safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$ . For example, the set of $10$ -safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$ . Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$ -safe, $11$ -safe, and $13$ -safe. | 958 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor a positive integer $p$ , define the positive integer $n$ to be $p$ -safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$ . For example, the set of $10$ -safe numbers is $\\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \\ldots\\}$ . Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$ -safe, $11$ -safe, and $13$ -safe.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A bee starts flying from point $P_0$ . She flies $1$ inch due east to point $P_1$ . For $j \ge 1$ , once the bee reaches point $P_j$ , she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$ . When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$ , where $a$ $b$ $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$
| 2,024 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA bee starts flying from point $P_0$ . She flies $1$ inch due east to point $P_1$ . For $j \\ge 1$ , once the bee reaches point $P_j$ , she turns $30^{\\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$ . When the bee reaches $P_{2015}$ she is exactly $a \\sqrt{b} + c \\sqrt{d}$ inches away from $P_0$ , where $a$ $b$ $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have? | 905 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $S$ be a subset of $\\{1,2,3,\\ldots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$ . What is the largest number of elements $S$ can have?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $A=\{1,2,3,4\}$ , and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$ . The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$ . | 453 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $A=\\{1,2,3,4\\}$ , and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$ . The probability that the range of $f$ and the range of $g$ are disjoint is $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$ , and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$ . What is the sum of the digits of $h_{2011}(1)$
| 16,089 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $f(x) = 10^{10x}, g(x) = \\log_{10}\\left(\\frac{x}{10}\\right), h_1(x) = g(f(x))$ , and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \\geq 2$ . What is the sum of the digits of $h_{2011}(1)$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
$\mathrm{(A) \ } \frac{\pi}{8} \qquad \mathrm{(B) \ } \frac{\pi}{7} \qquad \mathrm{(C) \ } \frac{\pi}{6} \qquad \mathrm{(D) \ } \frac{\pi}{5} \qquad \mathrm{(E) \ } \frac{\pi}{4}$ | 7 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTwo distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ radians is $180$ degrees.)\n$\\mathrm{(A) \\ } \\frac{\\pi}{8} \\qquad \\mathrm{(B) \\ } \\frac{\\pi}{7} \\qquad \\mathrm{(C) \\ } \\frac{\\pi}{6} \\qquad \\mathrm{(D) \\ } \\frac{\\pi}{5} \\qquad \\mathrm{(E) \\ } \\frac{\\pi}{4}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$ , and the system of equations
has exactly one solution. What is the minimum value of $c$
$\mathrm{(A)}\ 668 \qquad\mathrm{(B)}\ 669 \qquad\mathrm{(C)}\ 1002 \qquad\mathrm{(D)}\ 2003 \qquad\mathrm{(E)}\ 2004$ | 1,002 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nPositive integers $a,b,$ and $c$ are chosen so that $a<b<c$ , and the system of equations\nhas exactly one solution. What is the minimum value of $c$\n$\\mathrm{(A)}\\ 668 \\qquad\\mathrm{(B)}\\ 669 \\qquad\\mathrm{(C)}\\ 1002 \\qquad\\mathrm{(D)}\\ 2003 \\qquad\\mathrm{(E)}\\ 2004$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For positive real numbers $s$ , let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$ | 736 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor positive real numbers $s$ , let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$ $60$ , and $91$ . What is the fourth term of this sequence?
| 206 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$ $60$ , and $91$ . What is the fourth term of this sequence?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$ . A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$ , which is $7$ units from $\overline{BG}$ and $5$ units from $\overline{BC}$ . The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n}$ , where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n$ . | 230 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$ . A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$ , which is $7$ units from $\\overline{BG}$ and $5$ units from $\\overline{BC}$ . The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\\sqrt{n}$ , where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. | 29 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. \n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
$\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \text {(E)}\ 7$ | 5 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nOne morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?\n$\\text {(A)}\\ 3 \\qquad \\text {(B)}\\ 4 \\qquad \\text {(C)}\\ 5 \\qquad \\text {(D)}\\ 6 \\qquad \\text {(E)}\\ 7$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$ $e$ , and $10d+e$ , where $d$ and $e$ are single digits. What is the sum of the digits of $n$
$\mathrm{(A) \ } 12\qquad \mathrm{(B) \ } 15\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 24$ | 12 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$ $e$ , and $10d+e$ , where $d$ and $e$ are single digits. What is the sum of the digits of $n$\n$\\mathrm{(A) \\ } 12\\qquad \\mathrm{(B) \\ } 15\\qquad \\mathrm{(C) \\ } 18\\qquad \\mathrm{(D) \\ } 21\\qquad \\mathrm{(E) \\ } 24$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$ , it moves at random to one of the points $(a-1,b)$ , $(a,b-1)$ , or $(a-1,b-1)$ , each with probability $\tfrac{1}{3}$ , independently of its previous moves. The probability that it will hit the coordinate axes at $(0,0)$ is $\tfrac{m}{3^n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$ . Find $m + n$ . | 252 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$ , it moves at random to one of the points $(a-1,b)$ , $(a,b-1)$ , or $(a-1,b-1)$ , each with probability $\\tfrac{1}{3}$ , independently of its previous moves. The probability that it will hit the coordinate axes at $(0,0)$ is $\\tfrac{m}{3^n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$ . Find $m + n$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
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 distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt n}{p}$ , where $m$ $n$ , and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$ | 104 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \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 faces of the tetrahedron are all equal. This distance can be written in the form $\\frac{m \\sqrt n}{p}$ , where $m$ $n$ , and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$ . A plane passes through the midpoints of $AE$ $BC$ , and $CD$ . The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$ . Find $p$
AIME I 2007-13.png | 80 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$ . A plane passes through the midpoints of $AE$ $BC$ , and $CD$ . The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$ . Find $p$\nAIME I 2007-13.png\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$\mathrm{(A)\ }13\qquad\mathrm{(B)\ }14\qquad\mathrm{(C)\ }15\qquad\mathrm{(D)\ }16\qquad\mathrm{(E)\ }17$ | 13 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nMinneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?\n$\\mathrm{(A)\\ }13\\qquad\\mathrm{(B)\\ }14\\qquad\\mathrm{(C)\\ }15\\qquad\\mathrm{(D)\\ }16\\qquad\\mathrm{(E)\\ }17$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The sum of all positive integers $m$ such that $\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$ | 12 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe sum of all positive integers $m$ such that $\\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
$\mathrm{(A)}\ 20 \qquad \mathrm{(B)}\ 30 \qquad \mathrm{(C)}\ 55 \qquad \mathrm{(D)}\ 65 \qquad \mathrm{(E)}\ 80$ | 65 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAndrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?\n$\\mathrm{(A)}\\ 20 \\qquad \\mathrm{(B)}\\ 30 \\qquad \\mathrm{(C)}\\ 55 \\qquad \\mathrm{(D)}\\ 65 \\qquad \\mathrm{(E)}\\ 80$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior
of the octagon (not on the boundary) do two or more diagonals intersect?
| 49 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAll $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior\nof the octagon (not on the boundary) do two or more diagonals intersect?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Circle $C_1$ has its center $O$ lying on circle $C_2$ . The two circles meet at $X$ and $Y$ . Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$ $OZ=11$ , and $YZ=7$ . What is the radius of circle $C_1$
| 30 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nCircle $C_1$ has its center $O$ lying on circle $C_2$ . The two circles meet at $X$ and $Y$ . Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$ $OZ=11$ , and $YZ=7$ . What is the radius of circle $C_1$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Four lighthouses are located at points $A$ , $B$ , $C$ , and $D$ . The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$ , the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$ , and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$ . To an observer at $A$ , the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$ , the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac{p\sqrt{r}}{q}$ , where $p$ , $q$ , and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p+q+r$ . | 96 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFour lighthouses are located at points $A$ , $B$ , $C$ , and $D$ . The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$ , the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$ , and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$ . To an observer at $A$ , the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$ , the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\\frac{p\\sqrt{r}}{q}$ , where $p$ , $q$ , and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p+q+r$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
There 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 such that: | 902 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \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 such that:\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Triangle $ABC$ has $\angle BAC = 60^{\circ}$ $\angle CBA \leq 90^{\circ}$ $BC=1$ , and $AC \geq AB$ . Let $H$ $I$ , and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$ , respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$
| 80 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTriangle $ABC$ has $\\angle BAC = 60^{\\circ}$ $\\angle CBA \\leq 90^{\\circ}$ $BC=1$ , and $AC \\geq AB$ . Let $H$ $I$ , and $O$ be the orthocenter, incenter, and circumcenter of $\\triangle ABC$ , respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\\angle CBA$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$ , where $x$ is in $\mathcal{S}$ . In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$ . | 170 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}{1000}$ , where $x$ is in $\\mathcal{S}$ . In other words, $\\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\\mathcal{T}$ is divided by $1000$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The first four terms of an arithmetic sequence are $p$ $9$ $3p-q$ , and $3p+q$ . What is the $2010^\text{th}$ term of this sequence?
| 8,041 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe first four terms of an arithmetic sequence are $p$ $9$ $3p-q$ , and $3p+q$ . What is the $2010^\\text{th}$ term of this sequence?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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] |
For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$ , let $s(T)$ be the sum of the elements of $T$ , with $s(\emptyset)$ defined to be $0$ . If $T$ is chosen at random among all subsets of $U$ , the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$ | 1 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$ , let $s(T)$ be the sum of the elements of $T$ , with $s(\\emptyset)$ defined to be $0$ . If $T$ is chosen at random among all subsets of $U$ , the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$ , then Todd must say the next two numbers ( $2$ and $3$ ), then Tucker must say the next three numbers ( $4$ $5$ $6$ ), then Tadd must say the next four numbers ( $7$ $8$ $9$ $10$ ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$ th number said by Tadd?
| 5,979 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTravis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$ , then Todd must say the next two numbers ( $2$ and $3$ ), then Tucker must say the next three numbers ( $4$ $5$ $6$ ), then Tadd must say the next four numbers ( $7$ $8$ $9$ $10$ ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$ th number said by Tadd?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$ , if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , then \[a_{k+1} = \frac{m + 18}{n+19}.\] Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$ . | 59 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nConsider the sequence $(a_k)_{k\\ge 1}$ of positive rational numbers defined by $a_1 = \\frac{2020}{2021}$ and for $k\\ge 1$ , if $a_k = \\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , then \\[a_{k+1} = \\frac{m + 18}{n+19}.\\] Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\\frac{t}{t+1}$ for some positive integer $t$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$ , where $a$ and $b$ are positive real numbers. Find $a.$
| 118 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$ , where $a$ and $b$ are positive real numbers. Find $a.$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$ $M_3$ $M_5$ , and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$ $\overline{A_3 A_4}$ $\overline{A_5 A_6}$ , and $\overline{A_7 A_8}$ , respectively. For $i = 1, 3, 5, 7$ , ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$ $R_3 \perp R_5$ $R_5 \perp R_7$ , and $R_7 \perp R_1$ . Pairs of rays $R_1$ and $R_3$ $R_3$ and $R_5$ $R_5$ and $R_7$ , and $R_7$ and $R_1$ meet at $B_1$ $B_3$ $B_5$ $B_7$ respectively. If $B_1 B_3 = A_1 A_2$ , then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m + n$ | 37 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$ $M_3$ $M_5$ , and $M_7$ be the midpoints of sides $\\overline{A_1 A_2}$ $\\overline{A_3 A_4}$ $\\overline{A_5 A_6}$ , and $\\overline{A_7 A_8}$ , respectively. For $i = 1, 3, 5, 7$ , ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \\perp R_3$ $R_3 \\perp R_5$ $R_5 \\perp R_7$ , and $R_7 \\perp R_1$ . Pairs of rays $R_1$ and $R_3$ $R_3$ and $R_5$ $R_5$ and $R_7$ , and $R_7$ and $R_1$ meet at $B_1$ $B_3$ $B_5$ $B_7$ respectively. If $B_1 B_3 = A_1 A_2$ , then $\\cos 2 \\angle A_3 M_3 B_1$ can be written in the form $m - \\sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m + n$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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The function f is defined on the set of integers and satisfies $f(n)=\begin{cases} n-3&\mbox{if}\ n\ge 1000\\ f(f(n+5))&\mbox{if}\ n<1000\end{cases}$
Find $f(84)$ | 997 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe function f is defined on the set of integers and satisfies $f(n)=\\begin{cases} n-3&\\mbox{if}\\ n\\ge 1000\\\\ f(f(n+5))&\\mbox{if}\\ n<1000\\end{cases}$\nFind $f(84)$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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] |
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac {1}{3}$ . The probability that Club Truncator will finish the season with more wins than losses is $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | 341 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nClub Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$ . The probability that Club Truncator will finish the season with more wins than losses is $\\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
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What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which
\[k^2_1 + k^2_2 + ... + k^2_n = 2002?\]
| 17 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhat is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which\n\\[k^2_1 + k^2_2 + ... + k^2_n = 2002?\\]\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$ . What is the largest possible value of $n$
| 8 | [
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nArithmetic sequences $\\left(a_n\\right)$ and $\\left(b_n\\right)$ have integer terms with $a_1=b_1=1<a_2 \\le b_2$ and $a_n b_n = 2010$ for some $n$ . What is the largest possible value of $n$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
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] |
End of preview. Expand
in Data Studio
1750 challenging math QA combining historical AIME and AMC12, plus a subset of AMC10 filtered by pass@1 failure from Deekseek-Distill-Qwen-7B COT inference.
Each problem is prompted with both English and Chinese templates to enhance diversity for GRPO training.
Test set consists of open reference problem from AIMO1 and AIMO2 competitions.
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