A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.

Jan rolls a fair six-sided die and calls the result $r$. Then, he picks real numbers $a$ and $b$ between 0 and 1 uniformly at random and independently. If the probability that the polynomial $\tfrac{x^2}{r} - x\sqrt{a} + b$ has a real root can be expressed as simplified fraction $\frac{p}{q}$, find $p$. Let $T = TNYWR$. Compute the number of ordered triples $(a,b,c)$ such that $a$, $b$, and $c$ are distinct positive integers and $a + b + c = T$.

Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.

Recall that in any row of Pascal's Triangle, the first and last elements of the row are $1$ and each other element in the row is the sum of the two elements above it from the previous row. With this in mind, define the $\textit{Pascal Squared Triangle}$ as follows: [list] [*] In the $n^{\text{th}}$ row, where $n\geq 1$, the first and last elements of the row equal $n^2$; [*] Each other element is the sum of the two elements directly above it. [/list] The first few rows of the Pascal Squared Triangle are shown below. \[\begin{array}{c@{\hspace{7em}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c} \vspace{4pt} \text{Row 1: } & & & & & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 4 & & 4 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 9 & & 8 & & 9 & & & \\\vspace{4pt} \text{Row 4: } & & &16& &17& &17& & 16& & \\\vspace{4pt} \text{Row 5: } & &25 & &33& &34 & &33 & &25 & \end{array}\] Let $S_n$ denote the sum of the entries in the $n^{\text{th}}$ row. For how many integers $1\leq n\leq 10^6$ is $S_n$ divisible by $13$?

Let $ABC$ be a triangle. The incircle $\omega$ of $\triangle ABC$, which has radius $3$, is tangent to $\overline{BC}$ at $D$. Suppose the length of the altitude from $A$ to $\overline{BC}$ is $15$ and $BD^2 + CD^2 = 33$. What is $BC$?

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 $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. The tournament works as follows: First, all $\binom{36}{2}$ pairings of players are written down on slips of paper and are placed in a hat. Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match. We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes. What is the expected value of the number of matches played before the stopping point is reached? [i]Proposed by Dilhan Salgado[/i]

Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?

In an $n \times n$ square array of $1\times1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.

Let $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ be two squares such that the boundaries of $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ does not contain any line segment. Construct 16 line segments $A_iB_j$ for each possible $i,j \in \{1,2,3,4\}$. What is the maximum number of line segments that don't intersect the edges of $A_1A_2A_3A_4$ or $B_1B_2B_3B_4$? (intersection with a vertex is not counted). [i]Proposed by Allen Zheng[/i]

Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?

Given $n=2020$, sort the $6$ values $$n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}$$ from [b]least[/b] to [b]greatest[/b]. Give your answer as a 6 digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

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.

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.

Evaluate \[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\] [i]Proposed by Nathan Xiong[/i]