Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

Eric and Christina are playing a game with $n$ stones. They alternate taking some number of stones from the pile, with Eric going first. The number of stones Eric takes from the pile must be a power of $3$ (e.g. 1, 3, 9, 27, ...), while the number of stones Christina takes must be a power of $2$ (e.g. 1, 2, 4, 8, ...). Whoever takes the last stone wins. Find the sum of all $1\leq n \leq 100$ for which Eric has a winning strategy. [i]Proposed by Connor Gordon[/i]

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.

Let $ABC$ be a triangle with $AB = 5, BC = 13,$ and $AC = 12$. Let $D$ be a point on minor arc $AC$ of the circumcircle of $ABC$ (endpoints excluded) and $P$ on $\overline{BC}$. Let $B_1, C_1$ be the feet of perpendiculars from $P$ onto $CD, AB$ respectively and let $BB_1, CC_1$ hit $(ABC)$ again at $B_2, C_2$ respectively. Suppose that $D$ is chosen uniformly at random and $AD, BC, B_2C_2$ concur at a single point. Compute the expected value of $BP/PC$. [i]Proposed by Grant Yu[/i]

Compute the sum of all integers $x$ for which there exists an integer $y$ such that \[x^3+xy+y^3=503.\] [i]Proposed by Nathan Xiong[/i]

For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?

Find the number of ordered triples of positive integers $(a,b,c),$ where $1 \leq a,b,c \leq 10,$ with the property that $\gcd(a,b), \gcd(a,c),$ and $\gcd(b,c)$ are all pairwise relatively prime. [i]Proposed by Kyle Lee[/i]

Simplify $$\dbinom{2020}{1010}\dbinom{1010}{1010}+\dbinom{2019}{1010}\dbinom{1011}{1010}+\cdots+\dbinom{1011}{1010}\dbinom{2019}{1010} + \dbinom{1010}{1010}\dbinom{2020}{1010}.$$

In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

Five friends named Ella, Jacob, Muztaba, Peter, and William are suspicious of their friends for having secret group chats. Call a group of three people a "secret chat" if there is a chat with just the three of them (there cannot be multiple chats with the same three people). They have the following perfectly logical conversation in this order: [list] [*] Ella: I am part of $5$ secret chats. [*] Jacob: I know all of the secret chats that Ella is in. [*] Muztaba: Peter is in all but one of my secret chats. [*] Peter: I am in a secret chat that William cannot know exists. [*] William: I share exactly two secret chats with Jacob and two secret chats with Peter. [/list] Let $E$ be the number of chats Ella is in, $J$ the number of chats Jacob is in, $M$ the number of chats Muztaba is in, $P$ the number of chats Peter is in, and $W$ the number of chats William is in. Find $10000E$ $+$ $1000J$ $+$ $100M$ $+$ $10P+W$.

Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown: [asy] size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } [/asy] Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

Suppose you start at $0$, a friend starts at $6$, and another friend starts at $8$ on the number line. Every second, the leftmost person moves left with probability $\tfrac14$, the middle person with probability $\tfrac13$, and the rightmost person with probability $\tfrac12$. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.

Call a convex quadrilateral [i]angle-Pythagorean[/i] if the degree measures of its angles are integers $w\leq x \leq y \leq z$ satisfying $$w^2+x^2+y^2=z^2.$$ Determine the maximum possible value of $x+y$ for an angle-Pythagorean quadrilateral.

Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

There are distinct quadratics $e(x)$, $p(x)$, $h(x)$, $r(x)$, $a(x)$, and $m(x)$ with leading coefficient $1$, such that their roots are $2$ distinct values from the set $\{3, 4, 5, 6\}$. James takes three of these quadratics, sums two, and subtracts the last. Given that this new quadratic has a root at $0$, find its other root.

Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown: [asy] size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } [/asy] Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Andrew Wen[/i]

Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.

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

Let $A$, $F$, $L$, $M$, and $T$ be distinct digits such that $\overline{FALL} + \overline{LMT} = 2024$ and $F$, $L > 0$. Find the sum of all possible values of $\overline{FAT}$.

Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.

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 $\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}$ be two infinite sequences of real numbers, such that $\varepsilon_i \in \{-1,1\}$ for all $i$, and the numbers $\theta_i$ obey$$\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1$$and $\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}$. Compute the sum of all possible values of $$\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)$$ [i]Proposed by Grant Yu[/i]