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]

Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Nathan Xiong[/i]

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

Isabella assigns a distinct integer from $1$ to $6$ to each row and column of a $3\times 3$ grid. In each entry, she writes either the sum or the product of the values assigned to the corresponding row and column. Find the maximum possible value of the sum of all entries in the grid.

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]

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]

Nina and Meir are walking on a $3$ km path towards grandma's house. They start walking at the same time from the same point. Meir's speed is $4$ km/h and Nina's speed is $3$ km/h. Along the path there are several benches. Whenever Nina or Meir reaches a bench, they sit on it for some time and eat a cookie. Nina always takes $t$ minutes to eat a cookie, and Meir always takes $2t$ minutes to eat a cookie, where $t$ is a positive integer. It turns out that Nina and Meir reached grandma's house at the same time. How many benches were there? Find all of the options.

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]

Chris has a list of $5$ distinct numbers and every minute he independently and uniformly at random swaps a pair of them. Find the probability that after $4$ minutes the order of the list is the same as the original list.

An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute, along a straight road. If a governor prevents the speed of the car from exceeding $90$ miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least $6.6$ ft/sec.

Danyang is doing math. He starts to draw an isosceles triangle, but only manages to draws an angle of $70^{\circ}$ before he has to leave for recess. Find the sum of all possible values for the smallest angle in Danyang's triangle.

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. The common external tangent of the two circles closer to $X$ intersects $\omega_1$ and $\omega_2$ at $A$ and $B,$ respectively. Given that $AB=6,$ the radius of $\omega_1$ is $3,$ and $AY$ is tangent to $\omega_2,$ find $XY^2$.

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.)

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 $n$.

Find the sum of the distinct prime factors of $512512$.

[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]

Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.

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 $ABC.$

Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework? [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{7}$ Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games. [/hide]

A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. [i](Ahaan Rungta, 2 points)[/i]

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

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?

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? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) \equal{} 6$ have? [asy]size(220); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6); real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}; real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6}; draw(P1--P2--P3--P4--P5); dot("(-7, -4)",P1); dot("(-2, 6)",P2,LeftSide); dot("(1, 6)",P4); dot("(5, -6)",P5); xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6)); yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$