Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

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.

Let $ABCD$ be a unit square, and suppose that $E$ and $F$ are on $\overline{AD}$ and $\overline{AB}$ such that $AE = AF = \tfrac23$. Let $\overline{CE}$ and $\overline{DF}$ intersect at $G$. If the area of $\triangle CFG$ can be expressed as simplified fraction $\frac{p}{q}$, find $p + q$. Let $T = TNYWR$. A total of $2T$ students go on a road trip. They take two cars, each of which seats $T$ people. Call two students \textit{friendly} if they sat together in the same car going to the trip and in the same car going back home. What is the smallest possible number of friendly pairs of students on the trip?

$2$ identical red tokens and $2$ identical black tokens are placed on distinct cells of a $5\times5$ grid. Suppose it is impossible to color some additional cells of the grid red or black such that there exists a red path between the red tokens and a black path between the black tokens. Find the number of possible arrangements of the tokens on the grid. (A red path is a path of edge adjacent red cells, and same for a black path.)

Find the sum of all two-digit prime numbers whose digits are also both prime numbers. [i]Proposed by Nathan Xiong[/i]

Points $P$ and $Q$ lie on a circle $\omega$. The tangents to $\omega$ at $P$ and $Q$ intersect at point $T$, and point $R$ is chosen on $\omega$ so that $T$ and $R$ lie on opposite sides of $PQ$ and $\angle PQR = \angle PTQ$. Let $RT$ meet $\omega$ for the second time at point $S$. Given that $PQ = 12$ and $TR = 28$, determine $PS$.

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]

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.

Define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ to be $f(x)=(x+1)!-x!$. Find the number of positive integers $x<49$ such that $f(x)$ divides $f(49)$. [i]Proposed by David Tang[/i]

Suppose we have a regular $24$-gon labeled $A_1 \cdots A_{24}.$ We will draw $2$ similar $24$-gons within $A_1 \cdots A_{24}.$ For the sake of this problem, make $A_i=A_{i+24}.$ With our first configuration, we create $3$ stars by creating lines $\overline{A_iA_{i+9}}.$ A $24$-gon will be created in the center, which we denote as our first $24$-gon. With our second configuration, we create a start by creating lines $\overline{A_iA_{i+11}}.$ A $24$-gon will be created in the center, which we denote as our second $24$-gon. Find the ratio of the areas of the first $24$-gon to the second $24$-gon.

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 $mn$.

A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles? [i]Proposed by Giacomo Rizzo[/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]

Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

Determine, with proof, whether a square can be dissected into finitely many (not necessarily congruent) triangles, each of which has interior angles $30^\circ, 75^\circ,$ and $75^\circ.$

Albert writes down all of the multiples of $9$ between $9$ and $999,$ inclusive. Compute the sum of the digits he wrote.

Let $ABCD$ be a cyclic quadrilateral with $AB=3, BC=2, CD=6, DA=8,$ and circumcircle $\Gamma.$ The tangents to $\Gamma$ at $A$ and $C$ intersect at $P$ and the tangents to $\Gamma$ at $B$ and $D$ intersect at $Q.$ Suppose lines $PB$ and $PD$ intersect $\Gamma$ at points $W \neq B$ and $X \neq D,$ respectively. Similarly, suppose lines $QA$ and $QC$ intersect $\Gamma$ at points $Y \neq A$ and $Z \neq C,$ respectively. What is the value of $\frac{{WX}^2}{{YZ}^2}?$ [i]Proposed by Kyle Lee[/i]

David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?

Suppose integers $a < b < c$ satisfy \[ a + b + c = 95\qquad\text{and}\qquad a^2 + b^2 + c^2 = 3083.\] Find $c$.

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 $mn$.

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]

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