Found problems: 162
2025 AIME, 15
Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.
2020 AIME Problems, 9
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$
2025 AIME, 4
Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.
2001 AIME Problems, 14
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?
2020 AIME Problems, 13
Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
2025 AIME, 2
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$, as shown in the figure below.
[asy]
unitsize(14);
pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
filldraw(A--F--N--B--C--E--M--cycle, lightgray);
draw(A--B--C--cycle);
draw(D--M);
draw(N--G);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(M);
dot(N);
label("$A$", A, dir(90));
label("$B$", B, dir(225));
label("$C$", C, dir(315));
label("$D$", D, dir(135));
label("$E$", E, dir(135));
label("$F$", F, dir(45));
label("$G$", G, dir(45));
label("$M$", M, dir(45));
label("$N$", N, dir(135));
[/asy]
2023 AIME, 10
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 $x$.)
2015 AIME Problems, 12
Consider all 1000-element subsets of the set $\{1,2,3,\dots,2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2018 AIME Problems, 8
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 $d^2$.
CIME I 2018, 15
A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i]
2025 AIME, 10
The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle.
[asy]
unitsize(20);
add(grid(9,3));
draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2));
draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2));
real a = 0.5;
label("5",(a,a));
label("6",(1+a,a));
label("1",(2+a,a));
label("8",(3+a,a));
label("4",(4+a,a));
label("7",(5+a,a));
label("9",(6+a,a));
label("2",(7+a,a));
label("3",(8+a,a));
label("3",(a,1+a));
label("7",(1+a,1+a));
label("9",(2+a,1+a));
label("5",(3+a,1+a));
label("2",(4+a,1+a));
label("1",(5+a,1+a));
label("6",(6+a,1+a));
label("8",(7+a,1+a));
label("4",(8+a,1+a));
label("4",(a,2+a));
label("2",(1+a,2+a));
label("8",(2+a,2+a));
label("9",(3+a,2+a));
label("6",(4+a,2+a));
label("3",(5+a,2+a));
label("1",(6+a,2+a));
label("7",(7+a,2+a));
label("5",(8+a,2+a));
[/asy]
The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.
2021 AIME Problems, 13
Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.
2020 AIME Problems, 5
Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.
2017 AIME Problems, 12
Call a set $S$ [i]product-free[/i] if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
2015 AIME Problems, 11
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
2020 AIME Problems, 10
Let $m$ and $n$ be positive integers satisfying the conditions
[list]
[*] $\gcd(m+n,210) = 1,$
[*] $m^m$ is a multiple of $n^n,$ and
[*] $m$ is not a multiple of $n$.
[/list]
Find the least possible value of $m+n$.
2018 AIME Problems, 14
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.
2022 AIME Problems, 4
Let $w = \frac{\sqrt{3}+i}{2}$ and $z=\frac{-1+i\sqrt{3}}{2}$, where $i=\sqrt{-1}$. Find the number of ordered pairs $(r, s)$ of positive integers not exceeding $100$ that satisfy the equation $i\cdot w^r=z^s$.
CIME I 2018, 1
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
2022 AIME Problems, 15
Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
\begin{align*}
\sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\
\sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\
\sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}.
\end{align*}Then $\big[ (1-x)(1-y)(1-z) \big] ^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2018 AIME Problems, 2
The number \(n\) can be written in base \(14\) as \(\underline{a}\) \(\underline{b}\) \(\underline{c}\), can be written in base \(15\) as \(\underline{a}\) \(\underline{c}\) \(\underline{b}\), and can be written in base \(6\) as \(\underline{a}\) \(\underline{c}\) \(\underline{a}\) \(\underline{c}\), where \(a > 0\). Find the base-\(10\) representation of \(n\).
2025 AIME, 11
A piecewise linear function is defined by \[f(x) = \begin{cases} x & \text{if } x \in [-1, 1) \\ 2 - x & \text{if } x \in [1, 3)\end{cases}\] and $f(x + 4) = f(x)$ for all real numbers $x.$ The graph of $f(x)$ has the sawtooth pattern depicted below. [color=transparent]Diagram from RandomMath.[/color]
[center][img width=45]https://i.ibb.co/JW8jH2Dr/image.png[/img][/center]
The parabola $x = 34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-coordinates of these intersection points can be expressed in the form $\tfrac{a + b\sqrt c}d,$ where $a, b, c$ and $d$ are positive integers, $a, b,$ and $d$ has greatest common divisor equal to $1,$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$
2015 AIME Problems, 15
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
[asy]import three; import solids;
size(8cm);
currentprojection=orthographic(-1,-5,3);
picture lpic, rpic;
size(lpic,5cm);
draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight);
draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight);
draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight);
draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8));
draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed);
draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed);
draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0));
draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8));
size(rpic,5cm);
draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight);
draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed);
draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8));
draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8));
draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8));
label(rpic,"$A$",(-3,3*sqrt(3),0),W);
label(rpic,"$B$",(-3,-3*sqrt(3),0),W);
add(lpic.fit(),(0,0));
add(rpic.fit(),(1,0));[/asy]
2018 AIME Problems, 1
Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.
2016 AIME Problems, 8
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 $|m - n|$.