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circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
| 817
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\ncircle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ 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"
}
] |
Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$ . What is $\cos(\angle CMD)$
| 13
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $M$ be the midpoint of $\\overline{AB}$ in regular tetrahedron $ABCD$ . What is $\\cos(\\angle CMD)$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$ . Find the number of intersections of the graphs of \[y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).\]
| 385
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nDefine $f(x)=|| x|-\\tfrac{1}{2}|$ and $g(x)=|| x|-\\tfrac{1}{4}|$ . Find the number of intersections of the graphs of \\[y=4 g(f(\\sin (2 \\pi x))) \\quad\\text{ and }\\quad x=4 g(f(\\cos (3 \\pi y))).\\]\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\theta = \arg \left(\tfrac{w-z}{z}\right)$ . The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . (Note that $\arg(w)$ , for $w \neq 0$ , denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane)
| 100
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\\theta = \\arg \\left(\\tfrac{w-z}{z}\\right)$ . The maximum possible value of $\\tan^2 \\theta$ can be written as $\\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ . (Note that $\\arg(w)$ , for $w \\neq 0$ , denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane)\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$ , and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$ . What is the area of the circle that passes through vertices $A$ $B$ , and $C?$
| 26
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIsosceles triangle $ABC$ has $AB = AC = 3\\sqrt6$ , and a circle with radius $5\\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$ . What is the area of the circle that passes through vertices $A$ $B$ , and $C?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$ . Find the remainder when the smallest possible sum $m+n$ is divided by 1000.
| 371
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe domain of the function $f(x) = \\arcsin(\\log_{m}(nx))$ is a closed interval of length $\\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$ . Find the remainder when the smallest possible sum $m+n$ is divided by 1000.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$ $a$ $a$ $a$ $2a$ , and $b$ . What is the ratio of $b$ to $a$
| 3
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFour distinct points are arranged on a plane so that the segments connecting them have lengths $a$ $a$ $a$ $a$ $2a$ , and $b$ . What is the ratio of $b$ to $a$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
| 36
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \\times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For $\pi \le \theta < 2\pi$ , let \[P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots\] and \[Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots\] so that $\frac{P}{Q} = \frac{2\sqrt2}{7}$ . Then $\sin\theta = -\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
| 36
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor $\\pi \\le \\theta < 2\\pi$ , let \\[P=\\dfrac12\\cos\\theta-\\dfrac14\\sin2\\theta-\\dfrac18\\cos3\\theta+\\dfrac1{16}\\sin4\\theta+\\dfrac1{32}\\cos5\\theta-\\dfrac1{64}\\sin6\\theta-\\dfrac1{128}\\cos7\\theta+\\ldots\\] and \\[Q=1-\\dfrac12\\sin\\theta-\\dfrac14\\cos2\\theta+\\dfrac1{8}\\sin3\\theta+\\dfrac1{16}\\cos4\\theta-\\dfrac1{32}\\sin5\\theta-\\dfrac1{64}\\cos6\\theta+\\dfrac1{128}\\sin7\\theta +\\ldots\\] so that $\\frac{P}{Q} = \\frac{2\\sqrt2}{7}$ . Then $\\sin\\theta = -\\frac{m}{n}$ 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",
"role": "user"
}
] |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer
For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?
| 539
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer\nFor how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A strictly increasing sequence of positive integers $a_1$ $a_2$ $a_3$ $\cdots$ has the property that for every positive integer $k$ , the subsequence $a_{2k-1}$ $a_{2k}$ $a_{2k+1}$ is geometric and the subsequence $a_{2k}$ $a_{2k+1}$ $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$ . Find $a_1$
| 504
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA strictly increasing sequence of positive integers $a_1$ $a_2$ $a_3$ $\\cdots$ has the property that for every positive integer $k$ , the subsequence $a_{2k-1}$ $a_{2k}$ $a_{2k+1}$ is geometric and the subsequence $a_{2k}$ $a_{2k+1}$ $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$ . Find $a_1$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
| 13
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nJamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$
| 80
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn $\\triangle PAT,$ $\\angle P=36^{\\circ},$ $\\angle A=56^{\\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\\overline{TP}$ and $\\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\\overline{PA}$ and $\\overline{UG},$ respectively. What is the degree measure of the acute angle formed by lines $MN$ and $PA?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
| 28
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nEight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Consider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have length 14?
| 172
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nConsider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have length 14?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y,$ respectively. Suppose $PX=10,$ $PY=14,$ and $PQ=5.$ Then the area of trapezoid $XABY$ is $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n.$
| 33
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nCircles $\\omega_1$ and $\\omega_2$ intersect at two points $P$ and $Q,$ and their common tangent line closer to $P$ intersects $\\omega_1$ and $\\omega_2$ at points $A$ and $B,$ respectively. The line parallel to $AB$ that passes through $P$ intersects $\\omega_1$ and $\\omega_2$ for the second time at points $X$ and $Y,$ respectively. Suppose $PX=10,$ $PY=14,$ and $PQ=5.$ Then the area of trapezoid $XABY$ is $m\\sqrt{n},$ where $m$ and $n$ are positive 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"
}
] |
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8?
$\mathrm{(A)}\ 0\qquad \mathrm{(B)}\ 1\qquad \mathrm{(C)}\ 2\qquad \mathrm{(D)}\ 4\qquad \mathrm{(E)}\ 6$
| 4
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhat is the remainder when $3^0 + 3^1 + 3^2 + \\cdots + 3^{2009}$ is divided by 8?\n$\\mathrm{(A)}\\ 0\\qquad \\mathrm{(B)}\\ 1\\qquad \\mathrm{(C)}\\ 2\\qquad \\mathrm{(D)}\\ 4\\qquad \\mathrm{(E)}\\ 6$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$ . What is the remainder when $S$ is divided by $2017?$
| 1,024
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe number $2017$ is prime. Let $S = \\sum \\limits_{k=0}^{62} \\dbinom{2014}{k}$ . What is the remainder when $S$ is divided by $2017?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.$ How many dollars must LeRoy give to Bernardo so that they share the costs equally?
| 2
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.$ How many dollars must LeRoy give to Bernardo so that they share the costs equally?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer. For how many integer Fahrenheit temperatures between $32$ and $1000$ inclusive does the original temperature equal the final temperature?
| 539
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer. For how many integer Fahrenheit temperatures between $32$ and $1000$ inclusive does the original temperature equal the final temperature?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
| 4
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nConsider sequences of positive real numbers of the form $x, 2000, y, \\dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $d(n)$ denote the number of positive integers that divide $n$ , including $1$ and $n$ . For example, $d(1)=1,d(2)=2,$ and $d(12)=6$ . (This function is known as the divisor function.) Let \[f(n)=\frac{d(n)}{\sqrt [3]n}.\] There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$ . What is the sum of the digits of $N?$
| 9
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $d(n)$ denote the number of positive integers that divide $n$ , including $1$ and $n$ . For example, $d(1)=1,d(2)=2,$ and $d(12)=6$ . (This function is known as the divisor function.) Let \\[f(n)=\\frac{d(n)}{\\sqrt [3]n}.\\] There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\\ne N$ . What is the sum of the digits of $N?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$ , where $a$ $b$ $c$ , and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$ ? (Note that $i=\sqrt{-1}$
| 2
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$ , where $a$ $b$ $c$ , and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\\frac{-1+i\\sqrt{3}}{2} \\cdot r$ ? (Note that $i=\\sqrt{-1}$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$ . What are the coordinates of its new position?
| 5
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe point $(-1, -2)$ is rotated $270^{\\circ}$ counterclockwise about the point $(3, 1)$ . What are the coordinates of its new position?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation has more $1$ 's than $0$ 's. Find the remainder when $N$ is divided by $1000$
| 155
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $N$ be the number of positive integers that are less than or equal to $2003$ and whose base- $2$ representation has more $1$ 's than $0$ 's. Find the remainder when $N$ is divided by $1000$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying
| 349
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$
| 401
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nDefine $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.
| 72
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTwenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
$\mathrm{(A)}\ 30\% \qquad \mathrm{(B)}\ 40\% \qquad \mathrm{(C)}\ 49\% \qquad \mathrm{(D)}\ 51\% \qquad \mathrm{(E)}\ 70\%$
| 51
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAt Jefferson Summer Camp, $60\\%$ of the children play soccer, $30\\%$ of the children swim, and $40\\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?\n$\\mathrm{(A)}\\ 30\\% \\qquad \\mathrm{(B)}\\ 40\\% \\qquad \\mathrm{(C)}\\ 49\\% \\qquad \\mathrm{(D)}\\ 51\\% \\qquad \\mathrm{(E)}\\ 70\\%$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In rectangle $PQRS$ $PQ=8$ and $QR=6$ . Points $A$ and $B$ lie on $\overline{PQ}$ , points $C$ and $D$ lie on $\overline{QR}$ , points $E$ and $F$ lie on $\overline{RS}$ , and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$ , where $k$ $m$ , and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$
| 7
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn rectangle $PQRS$ $PQ=8$ and $QR=6$ . Points $A$ and $B$ lie on $\\overline{PQ}$ , points $C$ and $D$ lie on $\\overline{QR}$ , points $E$ and $F$ lie on $\\overline{RS}$ , and points $G$ and $H$ lie on $\\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\\sqrt{n}$ , where $k$ $m$ , and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$
| 375
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nComplex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many non- congruent triangles with perimeter $7$ have integer side lengths?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$
| 2
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many non- congruent triangles with perimeter $7$ have integer side lengths?\n$\\mathrm{(A) \\ } 1\\qquad \\mathrm{(B) \\ } 2\\qquad \\mathrm{(C) \\ } 3\\qquad \\mathrm{(D) \\ } 4\\qquad \\mathrm{(E) \\ } 5$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Two concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
| 13
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nTwo concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB=AC$ . The altitude from $A$ meets the opposite side at $D(-1, 3)$ . What are the coordinates of point $C$
| 49
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nPoints $A(11, 9)$ and $B(2, -3)$ are vertices of $\\triangle ABC$ with $AB=AC$ . The altitude from $A$ meets the opposite side at $D(-1, 3)$ . What are the coordinates of point $C$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
| 32
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?
| 465
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nNinety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively, and suppose that the splitting lines of $\triangle ABC$ through $M$ and $N$ intersect at $30^\circ.$ Find the perimeter of $\triangle ABC.$
| 459
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nGiven $\\triangle ABC$ and a point $P$ on one of its sides, call line $\\ell$ the $\\textit{splitting line}$ of $\\triangle ABC$ through $P$ if $\\ell$ passes through $P$ and divides $\\triangle ABC$ into two polygons of equal perimeter. Let $\\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\\overline{AB}$ and $\\overline{AC},$ respectively, and suppose that the splitting lines of $\\triangle ABC$ through $M$ and $N$ intersect at $30^\\circ.$ Find the perimeter of $\\triangle ABC.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For each even positive integer $x$ , let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square
| 899
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor each even positive integer $x$ , let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$
| 37
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p_{}$ can be written in the form $m/n$ 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"
}
] |
For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .
| 162
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$ . For that unique $a$ , find $a+U$ . (Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$ .)
| 944
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThere exists a unique positive integer $a$ for which the sum \\[U=\\sum_{n=1}^{2023}\\left\\lfloor\\dfrac{n^{2}-na}{5}\\right\\rfloor\\] is an integer strictly between $-1000$ and $1000$ . For that unique $a$ , find $a+U$ . (Note that $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x$ .)\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\]
| 1
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \\[(1+2a)(2+2b)(2a+b) = 32ab?\\]\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \le a_i \le 99$ . An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$ . Find $N$
| 202
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \\cdot 10^3 + a_2 \\cdot 10^2 + a_1 \\cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \\le a_i \\le 99$ . An example of such a representation is $1\\cdot 10^3 + 3\\cdot 10^2 + 67\\cdot 10^1 + 40\\cdot 10^0$ . Find $N$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\le A<B<C\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\le b<a<c<p$ , (c) $p$ divides $A-a$ , $B-b$ , and $C-c$ , and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .
| 272
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions: (a) $0\\le A<B<C\\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\\le b<a<c<p$ , (c) $p$ divides $A-a$ , $B-b$ , and $C-c$ , and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
There is a polynomial $P(x)$ with integer coefficients such that \[P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\] holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$ .
| 220
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThere is a polynomial $P(x)$ with integer coefficients such that \\[P(x)=\\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\\] holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
$ABCD$ is a square of side length $\sqrt{3} + 1$ . Point $P$ is on $\overline{AC}$ such that $AP = \sqrt{2}$ . The square region bounded by $ABCD$ is rotated $90^{\circ}$ counterclockwise with center $P$ , sweeping out a region whose area is $\frac{1}{c} (a \pi + b)$ , where $a$ $b$ , and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$ . What is $a + b + c$
| 19
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\n$ABCD$ is a square of side length $\\sqrt{3} + 1$ . Point $P$ is on $\\overline{AC}$ such that $AP = \\sqrt{2}$ . The square region bounded by $ABCD$ is rotated $90^{\\circ}$ counterclockwise with center $P$ , sweeping out a region whose area is $\\frac{1}{c} (a \\pi + b)$ , where $a$ $b$ , and $c$ are positive integers and $\\text{gcd}(a,b,c) = 1$ . What is $a + b + c$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p,$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$
| 134
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\\frac{-k+m\\sqrt{n}}p,$ where $k, m, n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The graphs of the equations
are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\tfrac{2}{\sqrt{3}}.\,$ How many such triangles are formed?
| 660
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe graphs of the equations\nare drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side length $\\tfrac{2}{\\sqrt{3}}.\\,$ How many such triangles are formed?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Given a point $P^{}_{}$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P_{}^{}$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\,$ and $\angle B=90^\circ.\,$ Then the area of the set of all fold points of $\triangle ABC\,$ can be written in the form $q\pi-r\sqrt{s},\,$ where $q, r,\,$ and $s\,$ are positive integers and $s\,$ is not divisible by the square of any prime. What is $q+r+s\,$
| 597
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nGiven a point $P^{}_{}$ on a triangular piece of paper $ABC,\\,$ consider the creases that are formed in the paper when $A, B,\\,$ and $C\\,$ are folded onto $P.\\,$ Let us call $P_{}^{}$ a fold point of $\\triangle ABC\\,$ if these creases, which number three unless $P^{}_{}$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,\\,$ and $\\angle B=90^\\circ.\\,$ Then the area of the set of all fold points of $\\triangle ABC\\,$ can be written in the form $q\\pi-r\\sqrt{s},\\,$ where $q, r,\\,$ and $s\\,$ are positive integers and $s\\,$ is not divisible by the square of any prime. What is $q+r+s\\,$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In parallelogram $ABCD$ , let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$ , and angle $ACB$ is $r$ times as large as angle $AOB$ . Find $\lfloor 1000r \rfloor$
| 777
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn parallelogram $ABCD$ , let $O$ be the intersection of diagonals $\\overline{AC}$ and $\\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$ , and angle $ACB$ is $r$ times as large as angle $AOB$ . Find $\\lfloor 1000r \\rfloor$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\] How many integers $n$ are there such that $P(n)\leq 0$
| 5,100
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nDefine \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
For all positive integers $n$ less than $2002$ , let
\begin{eqnarray*} a_n =\left\{ \begin{array}{lr} 11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\ 13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\ 14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\ 0, & \text{otherwise}. \end{array} \right. \end{eqnarray*}
Calculate $\sum_{n=1}^{2001} a_n$
$\mathrm{(A)}\ 448 \qquad\mathrm{(B)}\ 486 \qquad\mathrm{(C)}\ 1560 \qquad\mathrm{(D)}\ 2001 \qquad\mathrm{(E)}\ 2002$
| 448
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFor all positive integers $n$ less than $2002$ , let\n\\begin{eqnarray*} a_n =\\left\\{ \\begin{array}{lr} 11, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }13\\ \\text{and\\ }14;\\\\ 13, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }14\\ \\text{and\\ }11;\\\\ 14, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }11\\ \\text{and\\ }13;\\\\ 0, & \\text{otherwise}. \\end{array} \\right. \\end{eqnarray*}\nCalculate $\\sum_{n=1}^{2001} a_n$\n$\\mathrm{(A)}\\ 448 \\qquad\\mathrm{(B)}\\ 486 \\qquad\\mathrm{(C)}\\ 1560 \\qquad\\mathrm{(D)}\\ 2001 \\qquad\\mathrm{(E)}\\ 2002$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A list of $11$ positive integers has a mean of $10$ , a median of $9$ , and a unique mode of $8$ . What is the largest possible value of an integer in the list?
$\textbf {(A) } 24 \qquad \textbf {(B) } 30 \qquad \textbf {(C) } 31\qquad \textbf {(D) } 33 \qquad \textbf {(E) } 35$
| 35
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA list of $11$ positive integers has a mean of $10$ , a median of $9$ , and a unique mode of $8$ . What is the largest possible value of an integer in the list?\n$\\textbf {(A) } 24 \\qquad \\textbf {(B) } 30 \\qquad \\textbf {(C) } 31\\qquad \\textbf {(D) } 33 \\qquad \\textbf {(E) } 35$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$ .
| 797
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?
$\mathrm{(A)}\ 60\qquad \mathrm{(B)}\ 75\qquad \mathrm{(C)}\ 120\qquad \mathrm{(D)}\ 225\qquad \mathrm{(E)}\ 315$
| 315
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?\n$\\mathrm{(A)}\\ 60\\qquad \\mathrm{(B)}\\ 75\\qquad \\mathrm{(C)}\\ 120\\qquad \\mathrm{(D)}\\ 225\\qquad \\mathrm{(E)}\\ 315$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\overline{AB}$ parallel to $\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2 = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$
| 163
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nPolyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2 = p - q\\sqrt {r},$ where $p, q,$ and $r$ 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",
"role": "user"
}
] |
Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$ , and $DE=12$ . Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .
| 147
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$ , and $DE=12$ . Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
hexagon is inscribed in a circle . Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$ , has length $31$ . Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$
| 384
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nhexagon is inscribed in a circle . Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$ , has length $31$ . Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Mary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$
| 340
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nMary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\\ldots,\\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $\overline{AB}$ be a chord of a circle $\omega$ , and let $P$ be a point on the chord $\overline{AB}$ . Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$ . Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$ . Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$ . Line $PQ$ intersects $\omega$ at $X$ and $Y$ . Assume that $AP=5$ , $PB=3$ , $XY=11$ , and $PQ^2 = \tfrac{m}{n}$ , 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```\nLet $\\overline{AB}$ be a chord of a circle $\\omega$ , and let $P$ be a point on the chord $\\overline{AB}$ . Circle $\\omega_1$ passes through $A$ and $P$ and is internally tangent to $\\omega$ . Circle $\\omega_2$ passes through $B$ and $P$ and is internally tangent to $\\omega$ . Circles $\\omega_1$ and $\\omega_2$ intersect at points $P$ and $Q$ . Line $PQ$ intersects $\\omega$ at $X$ and $Y$ . Assume that $AP=5$ , $PB=3$ , $XY=11$ , and $PQ^2 = \\tfrac{m}{n}$ , 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"
}
] |
Convex pentagon $ABCDE$ has side lengths $AB=5$ , $BC=CD=DE=6$ , and $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$ .
| 60
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nConvex pentagon $ABCDE$ has side lengths $AB=5$ , $BC=CD=DE=6$ , and $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Assuming $a\neq3$ $b\neq4$ , and $c\neq5$ , what is the value in simplest form of the following expression? \[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]
| 1
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAssuming $a\\neq3$ $b\\neq4$ , and $c\\neq5$ , what is the value in simplest form of the following expression? \\[\\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\\]\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$ ?
| 130
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAn $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is the largest possible value of $p$ ?\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The number $2013$ is expressed in the form
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$
| 2
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe number $2013$ is expressed in the form\nwhere $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$
| 157
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nKathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy 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",
"role": "user"
}
] |
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$
| 56
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\\frac{z_3-z_1}{z_2-z_1}~\\cdot~\\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
| 25
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nBrianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Azar and Carl play a game of tic-tac-toe. Azar places an in $X$ one of the boxes in a $3$ -by- $3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$ . How many ways can the board look after the game is over?
| 148
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAzar and Carl play a game of tic-tac-toe. Azar places an in $X$ one of the boxes in a $3$ -by- $3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$ . How many ways can the board look after the game is over?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.
| 71
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is
$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35$
| 14
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe number $25^{64}\\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is\n$\\mathrm{(A) \\ } 7\\qquad \\mathrm{(B) \\ } 14\\qquad \\mathrm{(C) \\ } 21\\qquad \\mathrm{(D) \\ } 28\\qquad \\mathrm{(E) \\ } 35$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega.$ Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A,$ $\omega_B,$ and $\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
| 378
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nEquilateral triangle $\\triangle ABC$ is inscribed in circle $\\omega$ with radius $18.$ Circle $\\omega_A$ is tangent to sides $\\overline{AB}$ and $\\overline{AC}$ and is internally tangent to $\\omega.$ Circles $\\omega_B$ and $\\omega_C$ are defined analogously. Circles $\\omega_A,$ $\\omega_B,$ and $\\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\\sqrt{a} - \\sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
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 $m$ .
| 683
|
[
{
"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",
"role": "user"
}
] |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthling must occupy chair $15$ , Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N \cdot (5!)^3$ . Find $N$
| 346
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthling must occupy chair $15$ , Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N \\cdot (5!)^3$ . Find $N$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$
$\mathrm{(A)\ }5\qquad\mathrm{(B)\ }6\qquad\mathrm{(C)\ }7\qquad\mathrm{(D)\ }8\qquad\mathrm{(E)\ }9$
| 5
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nOn a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$\n$\\mathrm{(A)\\ }5\\qquad\\mathrm{(B)\\ }6\\qquad\\mathrm{(C)\\ }7\\qquad\\mathrm{(D)\\ }8\\qquad\\mathrm{(E)\\ }9$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$
| 201
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhat is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$ The area of $\mathcal{S}$ divided by the area of $\mathcal{T}$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n.$
| 25
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\\ge a, y\\ge b, z\\ge c.$ Let $\\mathcal{S}$ consist of those triples in $\\mathcal{T}$ that support $\\left(\\frac 12,\\frac 13,\\frac 16\\right).$ The area of $\\mathcal{S}$ divided by the area of $\\mathcal{T}$ is $m/n,$ 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 infinite product \[\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \cdots\] evaluates to a real number. What is that number?
| 10
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe infinite product \\[\\sqrt[3]{10} \\cdot \\sqrt[3]{\\sqrt[3]{10}} \\cdot \\sqrt[3]{\\sqrt[3]{\\sqrt[3]{10}}} \\cdots\\] evaluates to a real number. What is that number?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8$
| 5
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nMichael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?\n$\\mathrm{(A)}\\ 4\\qquad\\mathrm{(B)}\\ 5\\qquad\\mathrm{(C)}\\ 6\\qquad\\mathrm{(D)}\\ 7\\qquad\\mathrm{(E)}\\ 8$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four
times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
$\mathrm{(A)}\ 2500\qquad\mathrm{(B)}\ 3000\qquad\mathrm{(C)}\ 3500\qquad\mathrm{(D)}\ 4000\qquad\mathrm{(E)}\ 4500$
| 2,500
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nPostman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four \ntimes. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?\n$\\mathrm{(A)}\\ 2500\\qquad\\mathrm{(B)}\\ 3000\\qquad\\mathrm{(C)}\\ 3500\\qquad\\mathrm{(D)}\\ 4000\\qquad\\mathrm{(E)}\\ 4500$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$ $\omega_{B}$ to $BC$ and $BA$ $\omega_{C}$ to $CA$ and $CB$ , and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ . If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| 389
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFour circles $\\omega,$ $\\omega_{A},$ $\\omega_{B},$ and $\\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\\omega_{A}$ is tangent to sides $AB$ and $AC$ $\\omega_{B}$ to $BC$ and $BA$ $\\omega_{C}$ to $CA$ and $CB$ , and $\\omega$ is externally tangent to $\\omega_{A},$ $\\omega_{B},$ and $\\omega_{C}$ . If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\\omega$ can be represented in the form $\\frac{m}{n}$ , 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",
"role": "user"
}
] |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
| 127
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$ . Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$ . Let $Y$ be the intersection of lines $XG$ and $BC$ . Given that the measures of $\angle ABC, \angle BCA,$ and $\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$
| 592
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $\\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$ . Let $X$ be the intersection of the line tangent to the circumcircle of $\\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$ . Let $Y$ be the intersection of lines $XG$ and $BC$ . Given that the measures of $\\angle ABC, \\angle BCA,$ and $\\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\\angle BAC$ can be written as $\\frac{m}{n},$ 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",
"role": "user"
}
] |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$
| 407
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $m$ and $n$ be positive integers satisfying the conditions\n$\\quad\\bullet\\ \\gcd(m+n,210)=1,$\n$\\quad\\bullet\\ m^m$ is a multiple of $n^n,$ and\n$\\quad\\bullet\\ m$ is not a multiple of $n.$\nFind the least possible value of $m+n.$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$ , where $a$ $b$ , and $c$ are positive integers. Find $a+b+c$
| 98
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\\frac{\\sqrt[3]a + \\sqrt[3]b + 1}{c}$ , where $a$ $b$ , and $c$ are positive integers. Find $a+b+c$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$
| 849
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nAlpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$ , $x_1$ , $\dots$ , $x_{2011}$ such that \[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]
| 16
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the number of positive integers $m$ for which there exist nonnegative integers $x_0$ , $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$ . Find $100k$
| 86
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nSquare $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$ . Find $100k$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$
| 797
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
| 331
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nFind the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
In $\triangle ABC$ , we have $AB = 1$ and $AC = 2$ . Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$
$\mathrm{(A)}\ \frac{1+\sqrt{2}}{2} \qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2 \qquad\mathrm{(C)}\ \sqrt{2} \qquad\mathrm{(D)}\ \frac 32 \qquad\mathrm{(E)}\ \sqrt{3}$
| 2
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nIn $\\triangle ABC$ , we have $AB = 1$ and $AC = 2$ . Side $\\overline{BC}$ and the median from $A$ to $\\overline{BC}$ have the same length. What is $BC$\n$\\mathrm{(A)}\\ \\frac{1+\\sqrt{2}}{2} \\qquad\\mathrm{(B)}\\ \\frac{1+\\sqrt{3}}2 \\qquad\\mathrm{(C)}\\ \\sqrt{2} \\qquad\\mathrm{(D)}\\ \\frac 32 \\qquad\\mathrm{(E)}\\ \\sqrt{3}$\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
| 9
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nThe six-digit number $\\underline{2}\\,\\underline{0}\\,\\underline{2}\\,\\underline{1}\\,\\underline{0}\\,\\underline{A}$ is prime for only one digit $A.$ What is $A?$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
| 157
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nKathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy 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",
"role": "user"
}
] |
Of the following complex numbers $z$ , which one has the property that $z^5$ has the greatest real part?
| 3
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nOf the following complex numbers $z$ , which one has the property that $z^5$ has the greatest real part?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
How many nonnegative integers can be written in the form \[a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,\] where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$
| 3,281
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nHow many nonnegative integers can be written in the form \\[a_7\\cdot3^7+a_6\\cdot3^6+a_5\\cdot3^5+a_4\\cdot3^4+a_3\\cdot3^3+a_2\\cdot3^2+a_1\\cdot3^1+a_0\\cdot3^0,\\] where $a_i\\in \\{-1,0,1\\}$ for $0\\le i \\le 7$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
A clock chimes once at $30$ minutes past each hour and chimes on the hour according to the hour. For example, at $1 \text{PM}$ there is one chime and at noon and midnight there are twelve chimes. Starting at $11:15 \text{AM}$ on $\text{February 26, 2003},$ on what date will the $2003^{\text{rd}}$ chime occur?
| 9
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nA clock chimes once at $30$ minutes past each hour and chimes on the hour according to the hour. For example, at $1 \\text{PM}$ there is one chime and at noon and midnight there are twelve chimes. Starting at $11:15 \\text{AM}$ on $\\text{February 26, 2003},$ on what date will the $2003^{\\text{rd}}$ chime occur?\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$ , and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$ . Find the remainder when $m-n$ is divided by $1000$ .
| 0
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$ , and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$ . Find the remainder when $m-n$ is divided by $1000$ .\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
subset $B$ of the set of integers from $1$ to $100$ , inclusive, has the property that no two elements of $B$ sum to $125$ . What is the maximum possible number of elements in $B$
| 62
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nsubset $B$ of the set of integers from $1$ to $100$ , inclusive, has the property that no two elements of $B$ sum to $125$ . What is the maximum possible number of elements in $B$\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"
}
] |
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$
| 11
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nWhen $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$\n\n```\nReason step by step, and put your final answer within \\boxed{}. \n",
"role": "user"
}
] |
Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=120^\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
| 484
|
[
{
"content": "Solve the following AIME math problem. The answer should be an integer. \n```\nLet $ABCDE$ be a convex pentagon with $AB \\parallel CE, BC \\parallel AD, AC \\parallel DE, \\angle ABC=120^\\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ 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"
}
] |
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