Found problems: 634
2001 AIME Problems, 11
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 $m+n$.
2014 AIME Problems, 9
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
2024 AIME, 9
There is a collection of $25$ indistinguishable black chips and $25$ indistinguishable white chips. Find the number of ways to place some of these chips in $25$ unit cells of a $5 \times 5$ grid so that:
[list]
[*]each cell contains at most one chip,
[*]all chips in the same row and all chips in the same column have the same color,
[*]any additional chip placed on the grid would violate one or more of the previous two conditions.
[/list]
2025 AIME, 15
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
2024 AIME, 11
Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
2025 AIME, 9
The parabola with equation $y = x^2 - 4$ is rotated $60^\circ$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a - \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.
2013 AIME Problems, 10
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
2015 AIME Problems, 15
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
import cse5;
pathpen=black; pointpen=black;
size(6cm);
pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
D(CR((0,1),1)); D(CR((4,4),4,150,390));
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
D(MP("E",E,N));
[/asy]
2020 CHMMC Winter (2020-21), 8
Define
\[
S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1).
\]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2008 AIME Problems, 1
Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.
2009 AIME Problems, 15
Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \overline{MN}$ with the chords $ \overline{AC}$ and $ \overline{BC}$. The largest possible value of $ d$ can be written in the form $ r\minus{}s\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r\plus{}s\plus{}t$.
1983 AIME Problems, 6
Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.
2016 AIME Problems, 11
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P(\tfrac72)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2000 AIME Problems, 9
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$
2000 AIME Problems, 15
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
2002 AMC 12/AHSME, 24
Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$.
$ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$
2023 AIME, 14
A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$. The cube contains water whose surface is $7$ meters above $\mathcal P$. The volume of the water is $\tfrac mn$ cubic meters, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(250);
defaultpen(linewidth(0.6));
pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y;
pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W;
pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8);
filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2));
fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9));
draw(A--B--Z--X--A--Y--C--X^^C--D--Z);
draw(P1--P2--P3--P4--cycle^^D--P4);
dot("$A$",A,S);
dot("$B$",B,S);
dot("$C$",C,N);
dot("$D$",D,N);
label("$\mathcal P$",(-13,4.5));
[/asy]
1994 AIME Problems, 2
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
2009 AIME Problems, 10
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$.
2020 CHMMC Winter (2020-21), 2
Find the sum of all positive integers $x < 241$ such that both $x^{24} + x^{18} + x^{12} + x^6 + 1$ and $x^{20} + x^{10} + 1$ are multiples of $241$.
2021 AIME Problems, 14
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\triangle ABC$ 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 $m+n$.
[asy]
unitsize(5mm);
pair A,B,C,X,G,O,Y;
A = (2,8);
B = (0,0);
C = (15,0);
dot(A,5+black); dot(B,5+black); dot(C,5+black);
draw(A--B--C--A,linewidth(1.3));
draw(circumcircle(A,B,C));
O = circumcenter(A,B,C);
G = (A+B+C)/3;
dot(O,5+black); dot(G,5+black);
pair D = bisectorpoint(O,2*A-O);
pair E = bisectorpoint(O,2*G-O);
draw(A+(A-D)*6--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10));
draw(intersectionpoint(G--G+(G-E)*10,B--C)--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10));
X = intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10);
Y = intersectionpoint(G--G+(G-E)*10,B--C);
dot(Y,5+black);
dot(X,5+black);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$O$",O,ESE);
label("$G$",G,W);
label("$X$",X,dir(0));
label("$Y$",Y,NW);
draw(O--G--O--X--O--Y);
markscalefactor = 0.07;
draw(rightanglemark(X,G,O));
[/asy]
2024 AIME, 13
Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$.
2017 AIME Problems, 7
Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.
1999 AIME Problems, 14
Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$