Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them. Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.

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.

A kite with $AB = BC$ and $AD = CD$ has diagonals which satisfy $AC = 80$ and $BD = 71$. Let $AC$ and $BD$ intersect at a point $O$. Find the area of the quadrilateral formed by the circumcenters of $ABO$, $BCO$, $CDO$, and $ADO$.

Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.

Misha is currently taking a Complexity Theory exam, but he seems to have forgotten a lot of the material! In the question, he is asked to fill in the following boxes with $\subseteq$ and $\subsetneq$ to identify the relationship between different complexity classes: $$\mathsf{NL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{NP}\ \fbox{\phantom{tt}}\ \mathsf{PH}\ \fbox{\phantom{tt}}\ \mathsf{PSPACE}\ \fbox{\phantom{tt}}\ \mathsf {EXP}$$ and $$\mathsf{coNL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{coNP}\ \fbox{\phantom{tt}}\ \mathsf{PH}$$ Luckily, he remembers that $\mathsf{P} \neq \mathsf{EXP}$, $\mathsf{NL} \neq \mathsf{PSPACE}$, $\mathsf{coNL} \neq \mathsf{PSPACE}$, and $\mathsf{NP} \neq \mathsf{coNP}\implies \mathsf{P}\neq \mathsf{NP} \land \mathsf{P}\neq \mathsf{coNP}$. How many ways are there for him to fill in the boxes so as not to contradict what he remembers?

Let $\triangle ABC$ be equilateral. Let $D$ be the midpoint of side $AC,$ and let $DEFG$ be a square such that $D, F, B$ are collinear and $E,G$ lie on $AB,CB$ respectively. What fraction of the area of $\triangle ABC$ is covered by square $DEFG?$ [i]Proposed by Lohith Tummala[/i]

Adam places down cards one at a time from a standard 52 card deck (without replacement) in a pile. Each time he places a card, he gets points equal to the number of cards in a row immediately before his current card that are all the same suit as the current card. For instance, if there are currently two hearts on the top of the pile (and the third card in the pile is not hearts), then placing a heart would be worth 2 points, and placing a card of any other suit would be worth 0 points. What is the expected number of points Adam will have after placing all 52 cards? [i]Proposed by Adam Bertelli[/i]

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]

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 $m + n + p$.

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]

Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]

Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

In the national math league, there are $7$ teams. Their season is a round robin format, where each team plays other. Find the number of ways the games could go such that they have equal number of wins. [i]Proposed by Ishin Shah[/i]

Over all natural numbers $n$ with 16 (not necessarily distinct) prime divisors, one of them maximizes the value of $s(n)/n$, where $s(n)$ denotes the sum of the divisors of $n$. What is the value of $d(d(n))$, where $d(n)$ is the the number of divisors of $n$?

Let a tree on $mn + 1$ vertices be $(m,n)$-nice if the following conditions hold: [list] [*] $m + 1$ colors are assigned to the nodes of the tree [*] for the first $m$ colors, there will be exactly $n$ nodes of color $i$ $(1\le i \le m)$ [*] the root node of the tree will be the unique node of color $m+1$. \item the $(m,n)$-nice trees must also satisfy the condition that for any two non-root nodes $i, j$, if the color of $i$ equals the color of $j$, then the color of the parent of $i$ equals the color of the parent of $j$. [*] Nodes of the same color are considered indistinguishable (swapping any two of them results in the same tree). [/list] Let $N(u,v,l)$ denote the number of $(u,v)$-nice trees with $l$ leaves. Note that $N(2,2,2) = 2, N(2,2,3) = 4, N(2,2,4) = 6$. Compute the remainder when $\sum_{l = 123}^{789} N(8,101,l)$ is divided by $101$. Definition: Any rooted, ordered tree consists of some set of nodes, each of which has a (possibly empty) ordered list of children. Each node is the child of exactly one other node, with the exception of the root, which has not parent. There also cannot be any cycles of nodes which are all linearly children of each other. [i]Proposed by Advait Nene[/i]

For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$

Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$

Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\] Let $T = TNYWR$. $T$ people each put a distinct marble into a bag; its contents are mixed randomly and one marble is distributed back to each person. Given that at least one person got their own marble back, what is the probability that everyone else also received their own marble?

Compute the sum of all positive integers $x$ such that $(x-17)\sqrt{x-1}+(x-1)\sqrt{x+15}$ is an integer.

Find all sets of five positive integers whose mode, mean, median, and range are all equal to $5$.

A non-self intersecting hexagon $RANDOM$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points $$(0,0), (10,0), (10,10), (0,10), (3,4), (6,2).$$ Let $a_{\text{max}}$ be the greatest possible area of $RANDOM$ and $a_{\text{min}}$ the least possible area of $RANDOM.$ Find $a_{\text{max}}-a_{\text{min}}.$

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

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.)

Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.

Let $P$ be a point inside isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $$\angle{PAD}=\angle{PDA}=90^\circ-\angle{BPC}.$$ If $PA=14, AB=18,$ and $CD=28,$ compute the area of $ABCD.$