We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contains $111$. What is the length of the shortest string $s$ which contains the binary representations of all the positive integers less than or equal to $2048$?

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]

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]

Complex numbers $\omega_1, \ldots, \omega_n$ each have magnitude $1.$ Let $z$ be a complex number distinct from $\omega_1, \ldots, \omega_n$ such that $$\frac{z+\omega_1}{z-\omega_1}+\ldots+\frac{z+\omega_n}{z-\omega_n}=0.$$ Prove that $|z|=1.$

$1296$ CMU Students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the `score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students? [i]Proposed by Dilhan Salgado[/i]

Let $f:\mathbb{N}\to \mathbb{N}$ be a bijection satisfying $f(ab)=f(a)f(b)$ for all $a,b\in \mathbb{N}$. Determine the minimum possible value of $f(n)/n$, taken over all possible $f$ and all $n\leq 2019$.

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]

Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?

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$.)

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?

On a plane, two equilateral triangles (of side length $1$) share a side, and a circle is drawn with the common side as a diameter. Find the area of the set of all points that lie inside exactly one of these shapes. [i]Proposed by Howard Halim[/i]

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?

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 $100n + m$.

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 $10000E$ $+$ $1000J$ $+$ $100M$ $+$ $10P+W$.

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 $AB = AC = 10$ 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 $m + n$.

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]

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?

Equilateral triangle $T_0$ with side length $3$ is on a plane. Given triangle $T_n$ on the plane, triangle $T_{n+1}$ is constructed on the plane by translating $T_n$ by $1$ unit, in one of six directions parallel to one of the sides of $T_n$. The direction is chosen uniformly at random. Let $a$ be the least integer such that at most one point on the plane is in or on all of $T_0, T_1, T_2, \ldots, T_a$. It can be shown that $a$ exists with probability $1$. Find the probability that $a$ is even. [i]Proposed by Justin Hseih[/i]

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]

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?

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

There are $5$ people who start with $1, 2, 3, 4,$ and $5$ cookies, respectively. Every minute, two different people are chosen uniformly at random. If they have $a$ and $b$ cookies and $a\neq b,$ the person with more cookies eats $|a-b|$ of their own cookies. If $a = b,$ the minute still passes with nothing happening. Compute the expected number of minutes until all $5$ people have an equal number of cookies.

Suppose we have a cubic polynomial $p(x)$ such that $p(0)=0,p(1)=1,$ and $p(x)\leq \sqrt x$ for $0\leq x \leq 1.$ Suppose $p(0.5)$ is maximized. What is the sum of $p(0.25)+p(0.75)?$ [i]Proposed by Ishin Shah[/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.