Found problems: 634
2023 AIME, 6
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(2.5cm);
draw((0,0)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--cycle);
draw((0,1)--(1,1)--(1,0), dotted);
[/asy]
2021 AIME Problems, 4
There are real numbers $a, b, c, $ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i, $ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$
2024 AIME, 6
Consider the paths of length $16$ that go from the lower left corner to the upper right corner of an $8\times 8$ grid. Find the number of such paths that change direction exactly $4$ times.
2020 CHMMC Winter (2020-21), 4
Consider the minimum positive real number $\lambda$ such that for any two squares $A,B$ satisfying $\text{Area}(A) + \text{Area}(B)=1$, there always exists some rectangle $C$ of area $\lambda$, such that $A,B$ can be put inside $C$ and satisfy the following two constraints:
1. $A,B$ are non-overlapping;
2. the sides of $A$ and $B$ are parallel to some side of $C$.
$\lambda$ can be written as $\frac{\sqrt{m}+n}{p}$ for positive integers $m$, $n$, and $p$ where $n$ and $p$ are relatively prime. Find $m+n+p$.
2003 AIME Problems, 12
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?
2020 AIME Problems, 10
Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots+n^3$ is divided by $n+5$, the remainder is $17.$
2011 AIME Problems, 10
A circle with center $O$ has radius 25. Chord $\overline{AB}$ of length 30 and chord $\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder where $m+n$ is divided by 1000.
2014 AIME Problems, 12
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A) + \cos(3B) + \cos(3C) = 1$. Two sides of the triangle have lengths $10$ and $13$. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}$. Find $m$.
2014 AMC 12/AHSME, 19
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
2022 AIME Problems, 10
Find the remainder when $$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$ is divided by $1000$.
1996 AIME Problems, 6
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
1986 AIME Problems, 1
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?
2003 AIME Problems, 3
Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}$. Susan makes a list as follows: for each two-element subset of $\mathcal{S}$, she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
2018 AIME Problems, 11
Find the number of permutations of $1,2,3,4,5,6$ such that for each $k$ with $1\leq k\leq 5$, at least one of the first $k$ terms of the permutation is greater than $k$.
2016 AIME Problems, 7
For integers $a$ and $b$ consider the complex number \[\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a, b)$ such that this complex number is a real number.
2022 AIME Problems, 2
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2006 AIME Problems, 12
Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.
2010 AIME Problems, 10
Let $ N$ be the number of ways to write $ 2010$ in the form \[2010 \equal{} a_3 \cdot 10^3 \plus{} a_2 \cdot 10^2 \plus{} a_1 \cdot 10 \plus{} a_0,\] where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of such a representation is $ 1\cdot10^3 \plus{} 3\cdot10^2 \plus{} 67\cdot10^1 \plus{} 40\cdot10^0$. Find $ N$.
2015 AMC 10, 8
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2cm);
D((0,-2)--MP("y",(0,7),N));
D((-3,0)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
[/asy][asy]
import cse5;pathpen=black;pointpen=black;
unitsize(0.2cm);
D((0,-2)--MP("y",(0,7),N));
D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E));
D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5));
//
D((18,-2)--MP("y",(18,7),N));
D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E));
D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5));
//
D((36,-2)--MP("y",(36,7),N));
D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E));
D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3));
//
D((0,-17)--MP("y",(0,-8),N));
D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E));
D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10));
//
D((15,-17)--MP("y",(15,-8),N));
D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E));
D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15));
[/asy]
2011 AIME Problems, 11
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.
2020 AIME Problems, 8
A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of 5 units due east. Each night the bug rotates $60 ^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2020 CHMMC Winter (2020-21), 10
Let $\omega$ be a nonreal $47$th root of unity. Suppose that $\mathcal{S}$ is the set of polynomials of degree at most $46$ and coefficients equal to either $0$ or $1$. Let $N$ be the number of polynomials $Q \in \mathcal{S}$ such that
\[
\sum_{j = 0}^{46} \frac{Q(\omega^{2j}) - Q(\omega^{j})}{\omega^{4j} + \omega^{3j} + \omega^{2j} + \omega^j + 1} = 47.
\]
The prime factorization of $N$ is $p_1^{\alpha_1}p_2^{\alpha_2} \dots p_s^{\alpha_s}$ where $p_1, \ldots, p_s$ are distinct primes and $\alpha_1, \alpha_2, \ldots, \alpha_s$ are positive integers. Compute $\sum_{j = 1}^s p_j\alpha_j$.
2015 AIME Problems, 6
Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.[asy]
pair A,B,C,D,E,F,G,H,I,O;
O=(0,0);
C=dir(90);
B=dir(70);
A=dir(50);
D=dir(110);
E=dir(130);
draw(arc(O,1,50,130));
real x=2*sin(20*pi/180);
F=x*dir(228)+C;
G=x*dir(256)+C;
H=x*dir(284)+C;
I=x*dir(312)+C;
draw(arc(C,x,200,340));
label("$A$",A,dir(0));
label("$B$",B,dir(75));
label("$C$",C,dir(90));
label("$D$",D,dir(105));
label("$E$",E,dir(180));
label("$F$",F,dir(225));
label("$G$",G,dir(260));
label("$H$",H,dir(280));
label("$I$",I,dir(315));
[/asy]
2017 AIME Problems, 3
A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.