Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that \[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\]Prove that $\angle RAS+\angle BAC=\angle BGC$. Merlijn Staps

Let $n\ge m\ge 1$ be integers. Prove that \[\sum_{k=m}^n \left (\frac 1{k^2}+\frac 1{k^3}\right) \ge m\cdot \left(\sum_{k=m}^n \frac 1{k^2}\right)^2.\] Raymond Feng and Luke Robitaille

Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. Proposed by Merlijn Staps

Let $n\ge 3$ be an integer and let $K_n$ be the complete graph on $n$ vertices. Each edge of $K_n$ is colored either red, green, or blue. Let $A$ denote the number of triangles in $K_n$ with all edges of the same color, and let $B$ denote the number of triangles in $K_n$ with all edges of different colors. Prove \[ B\le 2A+\frac{n(n-1)}{3}.\](The complete graph on $n$ vertices is the graph on $n$ vertices with $\tbinom n2$ edges, with exactly one edge joining every pair of vertices. A triangle consists of the set of $\tbinom 32=3$ edges between $3$ of these $n$ vertices.) Proposed by Ankan Bhattacharya

Suppose $a,\,b,$ and $c$ are three complex numbers with product $1$. Assume that none of $a,\,b,$ and $c$ are real or have absolute value $1$. Define \begin{tabular}{c c c} $p=(a+b+c)+\left(\dfrac 1a+\dfrac 1b+\dfrac 1c\right)$ & \text{and} & $q=\dfrac ab+\dfrac bc+\dfrac ca$. \end{tabular}Given that both $p$ and $q$ are real numbers, find all possible values of the ordered pair $(p,q)$. David Altizio

Let $ABC$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle PAQ,\,\angle PBQ,$ and $\angle PCQ$ are the altitudes of triangle $ABC$. Prove that the midpoint of $\overline{PQ}$ lies on the Euler line of $ABC$. (The Euler line is the line through the circumcenter and orthocenter of a triangle.) Proposed by Holden Mui

The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera has a row of 2023 such coins alternately tails-up and heads-up, with the leftmost coin tails-up. In a move, Vera may flip over one of the coins in the row, subject to the following rules: On the first move, Vera may flip over any of the $2023$ coins. On all subsequent moves, Vera may only flip over a coin adjacent to the coin she flipped on the previous move. (We do not consider a coin to be adjacent to itself.) Determine the smallest possible number of moves Vera can make to reach a state in which every coin is heads-up. Luke Robitaille

Let $ABC$ be an equilateral triangle with side length $1$. Points $A_1$ and $A_2$ are chosen on side $BC$, points $B_1$ and $B_2$ are chosen on side $CA$, and points $C_1$ and $C_2$ are chosen on side $AB$ such that $BA_1<BA_2$, $CB_1<CB_2$, and $AC_1<AC_2$. Suppose that the three line segments $B_1C_2$, $C_1A_2$, $A_1B_2$ are concurrent, and the perimeters of triangles $AB_2C_1$, $BC_2A_1$, and $CA_2B_1$ are all equal. Find all possible values of this common perimeter. Ankan Bhattacharya

For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$th finite difference, denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by \begin{align*} \Delta^0 f(n)&=f(n)\\ \Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots. \end{align*}Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$. Holden Mui