Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is injective if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) Proposed by Evan Chen

Define a beautiful number to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. Proposed by Matthew Babbitt

We say a finite set $S$ of points in the plane is very if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an inversion with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) Proposed by Sammy Luo

Let $n$ be a positive integer and let $a_1$, $a_2$, \dots, $a_n$ be real numbers strictly between $0$ and $1$. For any subset $S$ of $\{1, 2, ..., n\}$, define \[ f(S) = \prod_{i \in S} a_i \cdot \prod_{j \not \in S} (1-a_j). \] Suppose that $\sum_{|S| \text{ odd}} f(S) = \frac{1}{2}$. Prove that $a_k = \frac{1}{2}$ for some $k$. (Here the sum ranges over all subsets of $\{1, 2, ..., n\}$ with an odd number of elements.) Proposed by Kevin Sun

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $\omega_1$ and $\omega_2$ denote the circumcircles of triangles $BOC$ and $BHC$, respectively. Suppose the circle with diameter $\overline{AO}$ intersects $\omega_1$ again at $M$, and line $AM$ intersects $\omega_1$ again at $X$. Similarly, suppose the circle with diameter $\overline{AH}$ intersects $\omega_2$ again at $N$, and line $AN$ intersects $\omega_2$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. Proposed by Sammy Luo

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? Proposed by David Yang