This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 634

2005 AIME Problems, 7
Tags: AMC , AIME
Let \[x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.\] Find $(x+1)^{48}$.

2023 AIME, 13
Tags: AMC , AIME , AIME I , ratio
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below. [asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]

2006 AIME Problems, 1
Tags: geometry , rectangle , AMC , AIME , 2006 AIME II
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.

1996 AIME Problems, 13
Tags: geometry , ratio , trigonometry , AMC , AIME , number theory , relatively prime
In triangle $ABC, AB=\sqrt{30}, AC=\sqrt{6},$ and $BC=\sqrt{15}.$ There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ and $\angle ADB$ is a right angle. The ratio \[ \frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)} \] can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2021 AIME Problems, 12
Tags: AMC , AIME , AIME II , trigonometry
A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2019 AIME Problems, 13
Tags: AMC , AIME , AIME II , geometry
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.

2022 AIME Problems, 11
Tags: AIME
Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$

2010 AIME Problems, 12
Tags: geometry , perimeter , ratio , AMC , AIME
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.

1995 AIME Problems, 8
Tags: AMC , AIME
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?

2018 AIME Problems, 6
Tags: complex numbers , AMC , AIME , AIME I , 2018 AIME I
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

2015 AIME Problems, 2
Tags: AMC , AIME , AIME I
The nine delegates to the Economic Cooperation Conference include 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2022 AIME Problems, 2
Tags: AMC , AIME , AIME I
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$, where $a$, $b$, and $c$ are (not necessarily distinct) digits.

2007 AIME Problems, 10
Tags: AIME , combinatorics
Let $S$ be a set with six elements. Let $P$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $P$. the probability that $B$ is contained in at least one of $A$ or $S-A$ is $\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)

2008 AIME Problems, 7
Tags: Vieta , AMC , AIME , algebra , polynomial , AIME I
Let $ r$, $ s$, and $ t$ be the three roots of the equation \[ 8x^3\plus{}1001x\plus{}2008\equal{}0.\]Find $ (r\plus{}s)^3\plus{}(s\plus{}t)^3\plus{}(t\plus{}r)^3$.

2022 AIME Problems, 6
Tags: AMC , AIME , AIME I
Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

1991 AIME Problems, 1
Tags: AMC , AIME , algebra , polynomial , Vieta
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]

2024 AIME, 10
Tags: geometry , AMC , AIME
Let $\triangle ABC$ have side lengths $AB = 5, BC = 9,$ and $CA = 10.$ The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D,$ and $\overline{AD}$ intersects the circumcircle at $P \ne A.$ The length of $\overline{AP}$ is equal to $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

2021 AIME Problems, 5
Tags: geometry , AMC , AIME , AIME II
For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.

2009 AMC 12/AHSME, 11
Tags: AMC , AIME , search , AMC 8
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? $ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$

2016 AIME Problems, 4
Tags: AMC , AIME , AIME II , Bernie Sanders
An $a\times b\times c$ rectangular box is built from $a\cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1\times b \times c$ parallel to the $(b\times c)$-faces of the box contains exactly $9$ red cubes, exactly 12 green cubes, and some yellow cubes. Each of the $b$ layers of size $a\times 1 \times c$ parallel to the $(a\times c)$-faces of the box contains exactly 20 green cubes, exactly 25 yellow cubes, and some red cubes. Find the smallest possible volume of the box.

2019 AIME Problems, 13
Tags: AIME , AIME I , geometry , circumcircle , 2019 AIME I , trigonometry , power of a point
Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

2023 AIME, 10
Tags: AMC , AIME
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$. [asy] size(160); defaultpen(linewidth(0.6)); for(int j=0;j<=6;j=j+1) { draw((j,0)--(j,2)); } for(int i=0;i<=2;i=i+1) { draw((0,i)--(6,i)); } for(int k=1;k<=12;k=k+1) { label("$"+((string) k)+"$",(floor((k-1)/2)+0.5,k%2+0.5)); } [/asy]

2020 CHMMC Winter (2020-21), 9
Tags: geometry , AIME , USAMO , IMO
Triangle $ABC$ has circumcenter $O$ and circumcircle $\omega$. Let $A_{\omega}$ be the point diametrically opposite $A$ on $\omega$, and let $H$ be the foot of the altitude from $A$ onto $BC$. Let $H_B$ and $H_C$ be the reflections of $H$ over $B$ and $C$, respectively. Point $P$ is the intersection of line $A_{\omega}B$ and the perpendicular of $BC$ at point $H_B$, and point $Q$ is the intersection of line $A_{\omega}C$ and the perpendicular of $CB$ at point $H_C$. The circles $\omega_1$ and $\omega_2$ have the respective centers $P$ and $Q$ and respective radii $PA$ and $QA$. Suppose that $\omega$, $\omega_1$, and $\omega_2$ intersect at another common point $X$. If $AO = \frac{\sqrt{105}}{5}$ and $AX = 4$, then $|AB - CA|^2$ can be written as $m - n\sqrt{p}$ for positive integers $m$ and $n$ and squarefree positive integer $p$. Find $m + n + p$. [i]Note: the reflection of a point $P$ over another point $Q \neq P$ is the point $P'$ such that $Q$ is the midpoint of $P$ and $P'$.[/i]

2011 AMC 12/AHSME, 24
Tags: geometry , AMC , AIME , trigonometry , cyclic quadrilateral
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? $ \textbf{(A)}\ \sqrt{15} \qquad\textbf{(B)}\ \sqrt{21} \qquad\textbf{(C)}\ 2\sqrt{6} \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 2\sqrt{7} $

2014 AIME Problems, 9
Tags: algebra , polynomial , Vieta , AMC , AIME , quadratics , trigonometry
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.