Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$

In an $n \times n$ square array of $1\times1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Compute the sum of all integers $x$ for which there exists an integer $y$ such that \[x^3+xy+y^3=503.\] [i]Proposed by Nathan Xiong[/i]

Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$? [asy] import olympiad; size(120); defaultpen(linewidth(0.8)); pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A; pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y); draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4")); draw(rightanglemark(A,B,C,3)); draw(A--B--C--cycle); draw((0,2)--(0,0)--(2,0),Arrows(size=8)); label("$6$",(PAx+PBx)/2,S); label("$8$",(PAy+PCy)/2,W); [/asy]

Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?

Compute the number of ordered triples $(x,y,z)$ of integers satisfying \[x^2+y^2+z^2=9.\] [i]Proposed by Nathan Xiong[/i]

Let $f(x)=x^4-4x^2+2.$ Find the smallest natural $n \in \mathbb{N}$ such that there exists $k,c \in \mathbb{N}$ with $$\left|f^k\left(\frac{n^2+1}{n}\right)-c^{144}\right| < \frac{1}{100}.$$

The integers from $1$ to $9$ are arranged in a $3\times3$ grid. The rows and columns of the grid correspond to $6$ three-digit numbers, reading rows from left to right, and columns from top to bottom. Compute the least possible value of the largest of the $6$ numbers.

Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the $\textit{Pascal Squared Triangle}$ as follows: [list] [*] In the $n^{\text{th}}$ row, where $n\geq 1$, the first and last elements of the row equal $n^2$; [*] Each other element is the sum of the two elements directly above it. [/list] The first few rows of the Pascal Squared Triangle are shown below. \[\begin{array}{c@{\hspace{7em}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c} \vspace{4pt} \text{Row 1: } & & & & & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 4 & & 4 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\\vspace{4pt} \text{Row 4: } & & &16& &17& &17& & 16& & \\\vspace{4pt} \text{Row 5: } & &25 & &33& &34 & &33 & &25 & \end{array}\] Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1\leq n\leq 10^6$ is $S_n$ divisible by $13$?

Define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ to be $f(x)=(x+1)!-x!$. Find the number of positive integers $x<49$ such that $f(x)$ divides $f(49)$. [i]Proposed by David Tang[/i]

In a $2024 \times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^\circ$ clockwise or counterclockwise if it is currently on a black square, otherwise it must continue walking in the same direction. A coloring of the grid is called [i]simple[/i] if it is [b]not[/b] possible for the ant to arrive back at its starting location after some time. How many simple colorings of the grid are maximal, in the sense that adding any black square results in a coloring that is not simple?

For how many permutations $\pi$ of $\{1,2,\ldots,9\}$ does there exist an integer $N$ such that \[N\equiv \pi(i)\pmod{i}\text{ for all integers }1\leq i\leq 9?\]

Brandon wants to split his orchestra of $20$ violins, $15$ violas, $10$ cellos, and $5$ basses into three distinguishable groups, where all of the players of each instrument are indistinguishable. He wants each group to have at least one of each instrument and for each group to have more violins than violas, more violas than cellos, and more cellos than basses. How many ways are there for Brandon to split his orchestra following these conditions?

In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

Suppose you start at $0$, a friend starts at $6$, and another friend starts at $8$ on the number line. Every second, the leftmost person moves left with probability $\tfrac14$, the middle person with probability $\tfrac13$, and the rightmost person with probability $\tfrac12$. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.

For two real numbers $x$ and $y$, let $x\circ y=\frac{xy}{x+y}$. The value of \[1 \circ (2 \circ (3 \circ (4 \circ 5)))\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

A [i]weird checkerboard[/i] is a coloring of an $8\times8$ grid constructed by making some (possibly none or all) of the following $14$ cuts: [list] [*] the $7$ vertical cuts along a gridline through the entire height of the board, [*] and the $7$ horizontal cuts along a gridline through the entire width of the board. [/list] The divided rectangles are then colored black and white such that the bottom left corner of the grid is black, and no two rectangles adjacent by an edge share a color. Compute the number of weird checkerboards that have an equal amount of area colored black and white. [center] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f768a7a51c9c9bc56a1d55427c33e15e4bcd74.png[/img] [/center]

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

Let $ABCD$ be an isosceles trapezoid with $2DA=2AB=2BC=CD$. A point $P$ lies in the interior of $ABCD$ such that $BP=1$, $CP=2$, $DP=4$. Find the area of $ABCD$.

Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$

Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.

There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. The tournament works as follows: First, all $\binom{36}{2}$ pairings of players are written down on slips of paper and are placed in a hat. Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match. We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes. What is the expected value of the number of matches played before the stopping point is reached? [i]Proposed by Dilhan Salgado[/i]

We are searching for the number $7$ in the following binary tree: [center] [img] https://cdn.artofproblemsolving.com/attachments/8/c/70ad159d239e9fd8dd9775e6391965e1016f03.png [/img] [/center] We use the following algorithm (which terminates with probability $1$): [list=1] [*] Write down the number currently at the root node. [*] If we wrote down $7,$ terminate. [*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge [*] Go back to step $1.$ [/list] Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$