Let $ZHAO$ be a square with area $2024$. Let $X$ be the center of this square and let $C$, $D$, $E$, $K$ be the centroids of $XZH$, $XHA$, $XAO$, and $XOZ$, respectively. Find $[ZHAO]$ $+$ $[CZHAO]$ $+$ $[DZHAO]$ $+$ $[EZHAO]$ $+$ $[KZHAO]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)

In the center of a square is a rabbit and at each vertex of this even square, a wolf. The wolves only move along the sides of the square and the rabbit moves freely in the plane. Knowing that the rabbit move at a speed of $10$ km / h and that the wolves move to a maximum speed of $14$ km / h, determine if there is a strategy for the rabbit to leave the square without being caught by the wolves.

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

[asy] size(5cm); defaultpen(fontsize(6pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle); draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle); draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle); draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2); label("1", (-3.5,0), S); label("2", (-2,0), S); label("1", (-0.5,0), S); label("1", (3.5,0), S); label("2", (2,0), S); label("1", (0.5,0), S); label("1", (0,3.5), E); label("2", (0,2), E); label("1", (0,0.5), E); label("1", (0,-3.5), E); label("2", (0,-2), E); label("1", (0,-0.5), E); [/asy] Compute the area of the resulting shape, drawn in red above. [i]Proposed by Nathan Xiong[/i]

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$. [i]Proposed by Muztaba Syed[/i] [hide=Solution] [i]Solution.[/i] $\boxed{9}$ $a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$. [/hide]

Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$. [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{4}$ We have $x+2*2+y=x \cdot 2+2 \cdot y$. When simplifying, we have $x+y+4=2x+2y$, and $x+y=4$. [/hide]

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Henry places some rooks and some kings in distinct cells of a $2\times 8$ grid such that no two rooks attack each other and no two kings attack each other. Find the maximum possible number of pieces on the board. (Two rooks [i]attack[/i] each other if they are in the same row or column and no pieces are between them. Two kings attack each other if their cells share a vertex.)

Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andy Xu[/i]

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the feet from $B$ and $C$ to the median from $A$, respectively. Suppose $DE=4$ and $CD=5$. Find the area of $ABC.$

Let $P(n)$ denote the product of digits of $n$. Find the number of positive integers $n \leq 2024$ where $P(n)$ is divisible by $n$.

The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even. [i]Proposed by Muztaba Syed and Derek Zhao[/i] [hide=Solution] [i]Solution. [/i]$\boxed{\dfrac{1}{3}}$ Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the answer is $\boxed{\dfrac{1}{3}}$.[/hide]

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

What is $2021+20+21+2+0+2+1$? [i]Proposed by Nathan Xiong[/i]

The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even. [i]Proposed by Muztaba Syed and Derek Zhao[/i] [hide=Solution] [i]Solution. [/i]$\boxed{\dfrac{1}{3}}$ Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the answer is $\boxed{\dfrac{1}{3}}$.[/hide]

Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$. [i]Proposed by Andrew Wen[/i]