Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.

Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown: [asy] size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } [/asy] Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

Compute \[\frac{(1-\tan10^\circ)(1-\tan 20^\circ)(1-\tan30^\circ)(1-\tan40^\circ)}{(1-\tan5^\circ)(1-\tan 15^\circ)(1-\tan25^\circ)(1-\tan35^\circ)}.\] [i]Proposed by Connor Gordon[/i]

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250.

For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even? [i]Proposed by Andrew Wen[/i]

A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.

Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$. Let $T = TNYWR$. For some positive integer $k$, a circle is drawn tangent to the coordinate axes such that the lines $x + y = k^2, x + y = (k+1)^2, \dots, x+y = (k+T)^2$ all pass through it. What is the minimum possible value of $k$?

Determine the maximum possible value of $$\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})$$ over all $x\in [0,1]$.

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]

Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?

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]

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

Solve for $x$ if $\sqrt{x + 1}+ \sqrt{x} = 5.$ [i]Proposed by Eric Oh[/i]

Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find \[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? [i]Proposed by Nathan Xiong[/i]

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/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 $$p(1)+p(2)+p(3)+p(5)+p(6).$$

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 $mn$.

Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

Five friends named Ella, Jacob, Muztaba, Peter, and William are suspicious of their friends for having secret group chats. Call a group of three people a "secret chat" if there is a chat with just the three of them (there cannot be multiple chats with the same three people). They have the following perfectly logical conversation in this order: [list] [*] Ella: I am part of $5$ secret chats. [*] Jacob: I know all of the secret chats that Ella is in. [*] Muztaba: Peter is in all but one of my secret chats. [*] Peter: I am in a secret chat that William cannot know exists. [*] William: I share exactly two secret chats with Jacob and two secret chats with Peter. [/list] Let $E$ be the number of chats Ella is in, $J$ the number of chats Jacob is in, $M$ the number of chats Muztaba is in, $P$ the number of chats Peter is in, and $W$ the number of chats William is in. Find $10000E$ $+$ $1000J$ $+$ $100M$ $+$ $10P+W$.

There are $9$ points arranged in a $3\times 3$ square grid. Let two points be adjacent if the distance between them is half the side length of the grid. (There should be $12$ pairs of adjacent points). Suppose that we wanted to connect $8$ pairs of adjacent points, such that all points are connected to each other. In how many ways is this possible? [i]Proposed by Kevin You[/i]