Find the area of the quadrilateral with vertices at $(0,0), (2,0), (20,24), (0,2)$ in that order.

$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

[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]

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/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]

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]

Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$. [i]Proposed by Nathan Xiong[/i]

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]

If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$. $$xy+yz = 30$$ $$yz+zx = 36$$ $$zx+xy = 42$$ [i]Proposed by Nathan Xiong[/i]

Call a number [i]orz[/i] if it is a positive integer less than $2024$. Call a number [i]admitting[/i] if it can be expressed as $a^2-1$ where $a$ is a positive integer. Finally call a number [i]muztaba[/i] if it has exactly $4$ positive integer factors. Find the number of [i]muztaba admitting orz[/i] numbers.

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.

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]

A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.

A train passes an observer in $t_1$ sec. At the same speed the train crosses a bridge $\ell$ m long. It takes the train $t_2$ sec to cross the bridge from the moment the locomotive drives onto the bridge until the last car leaves it. Find the length and speed of the train.

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]

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 $a_n$ be a sequence such that $a_1=1$, $a_2=1$, and $a_{n+2}=\tfrac{a_{n+1}a_n}{a_{n+1}+a_n}$. Find the value of \[\sum_{n=1}^\infty \frac{1}{a_n3^n}.\]

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 $ABCD$ 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]

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

Find the number of positive integers $x$ that satisfy \[ \left \lfloor{\frac{2024}{ \left \lfloor \frac{2024}{x} \right \rfloor }} \right \rfloor = x.\]

Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$. [i]Proposed by Nathan Xiong[/i]

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 $\mathcal P$.)

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]