Found problems: 138
2024 AIME, 6
Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$. Bob's list has $2024$ sets. Find the sum of the elements of $A$
2022 AIME Problems, 12
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that $$\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.$$ Find the least possible value of $a+b.$
2017 AIME Problems, 13
For each integer $n\ge 3$, let $f(n)$ be the number of 3-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.
2007 AIME Problems, 6
An integer is called [i]parity-monotonic[/i] if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ is $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
2019 AIME Problems, 12
For $n \ge 1$ call a finite sequence $(a_1, a_2 \ldots a_n)$ of positive integers [i]progressive[/i] if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \le i \le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.
2022 AIME Problems, 5
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.
2022 AIME Problems, 1
Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.
2024 AIME, 7
Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.
2023 AIME, 3
Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$.
2023 AIME, 9
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 line $AB$ 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 $m + n$.
CIME II 2018, 12
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.
[I]Proposed by [b]FedeX333X[/b][/I]
2021 AIME Problems, 2
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.
[asy]
pair A,B,C,D,E,F,G;
B=origin;
A=5*dir(60);
C=(5,0);
E=0.6*A+0.4*B;
F=0.6*A+0.4*C;
G=rotate(240,F)*A;
D=extension(E,F,B,dir(90));
draw(D--G--A,grey);
draw(B--0.5*A+rotate(60,B)*A*0.5,grey);
draw(A--B--C--cycle,linewidth(1.5));
dot(A^^B^^C^^D^^E^^F^^G);
label("$A$",A,dir(90));
label("$B$",B,dir(225));
label("$C$",C,dir(-45));
label("$D$",D,dir(180));
label("$E$",E,dir(-45));
label("$F$",F,dir(225));
label("$G$",G,dir(0));
label("$\ell$",midpoint(E--F),dir(90));
[/asy]
2019 AIME Problems, 15
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
2008 AIME Problems, 15
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
2024 AIME, 2
A list of positive integers has the following properties:
- The sum of the items in the list is $30$.
- The unique mode of the list is $9$.
- The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
CIME II 2018, 5
Laurie plays a game called $\textit{bash}$ where she picks two distinct numbers between $1$ and $10$, inclusive, at random. She then finds their sum, product, and non-negative difference. At random, she picks two of these three numbers and tells them to Ali. If the probability that Ali is able to logically deduce the original numbers can be written as $\frac{m}{n}$, with $m$ and $n$ relatively prime, find $m+n$.
[i]Proposed by [b] atmchallenge [/b][/i]
2018 AIME Problems, 9
Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC= DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\times 27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{HA}$, and partition the octagon into $7$ triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.
[asy]
unitsize(6);
pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0);
pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17);
draw(P--Q--R--SS--cycle);
draw(J--B);
draw(J--C);
draw(J--D);
draw(J--EE);
draw(J--F);
draw(J--G);
draw(A--B);
draw(H--G);
real dark = 0.6;
filldraw(A--B--P--cycle, gray(dark));
filldraw(H--G--Q--cycle, gray(dark));
filldraw(F--EE--R--cycle, gray(dark));
filldraw(D--C--SS--cycle, gray(dark));
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE);
dot(F);
dot(G);
dot(H);
dot(J);
dot(H);
defaultpen(fontsize(10pt));
real r = 1.3;
label("$A$", A, W*r);
label("$B$", B, S*r);
label("$C$", C, S*r);
label("$D$", D, E*r);
label("$E$", EE, E*r);
label("$F$", F, N*r);
label("$G$", G, N*r);
label("$H$", H, W*r);
label("$J$", J, W*r);
[/asy]
2025 AIME, 1
Six points $A, B, C, D, E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC=26, BD=22, CE=31, DF=33, AF=73, CG=40,$ and $DG=30.$ Find the area of $\triangle BGE.$
2025 AIME, 9
There are $n$ values of $x$ in the interval $0<x<2\pi$ where $f(x)=\sin(7\pi\cdot\sin(5x))=0$. For $t$ of these $n$ values of $x$, the graph of $y=f(x)$ is tangent to the $x$-axis. Find $n+t$.
CIME II 2018, 8
Triangle $ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. The internal angle bisector of $\angle ABC$ intersects side $CA$ at $X$. The circumcircles of triangles $AXB$ and $BXC$ intersect sides $BC$ and $AB$ at $M$ and $N$, respectively. The value of $MN^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m+n$ is divided by $1000$.
[i]Proposed by [b] Th3Numb3rThr33[/b][/i]
2016 AIME Problems, 14
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.
CIME II 2018, 15
Anne and Bill decide to play a game together. At the beginning, they chose a positive integer $n$; then, starting from a positive integer $\mathcal{N}_0$, Anne subtracts to $\mathcal{N}_0$ an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1=\mathcal{N}_0-n^k$ is then passed to Bill, who repeats the same process starting from $\mathcal{N}_1$: he subtracts to $\mathcal{N}_1$ an integer $j$-th power of $n$ less than or equal to $\mathcal{N}_1$, and he then gives the resulting number $\mathcal{N}_2=\mathcal{N}_1-n^j$ to Anne. The game continues like that until one player gets $0$ as the result of his operation, winning the game. For each $1\leq n \leq 1000$, let $f(n)$ be the number of integers $1\leq \mathcal{N}_0\leq 5000$ such that Anne has a winning strategy starting from them. For how many values of $n$ we have that $f(n)\geq 2520$?
[I]Proposed by [b]FedeX333X[/b][/I]
2018 AIME Problems, 12
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 $\triangle APB$ and $\triangle CPD$ equals the sum of the areas of $\triangle BPC$ and $\triangle APD$. Find the area of quadrilateral $ABCD$.
2007 AIME Problems, 2
Find the number of ordered triple $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$.
2017 AIME Problems, 15
Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, define $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.