There are $a+b$ bowls in a row, numbered $1$ through $a+b$, where $a,b$ are positive integers. Initially the first $a$ bowls each hold an apple and the last $b$ each hold a pear. A legal move moves an apple from bowl $i$ to $i+1$ and a pear from bowl $j$ to $j-1$, provided $i-j$ is even. Multiple fruits may share a bowl. Prove that the first $b$ bowls can end with one pear each and the last $a$ with one apple each if and only if $ab$ is even.
Proposed by Jim Propp. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p12189456.
Let $ABCD$ be a cyclic quadrilateral with $AD^2+BC^2=AB^2$. Its diagonals meet at $E$. A point $P$ on side $AB$ satisfies $\angle APD=\angle BPC$. Prove that line $PE$ passes through the midpoint of $CD$.
Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party solution notes.
Let $ABC$ be a triangle with $\angle B>90^\circ$, and let $E,F$ be the feet of the altitudes from $B,C$, respectively. Can line $EF$ be tangent to the $A$-excircle?
Proposed by Ankan Bhattacharya, Zack Chroman, Anant Mudgal. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p12195848.
Let $n\ge0$ be an integer. Count families of sets $S_{ij}\subseteq\{1,\ldots,2n\}$ for $0\le i,j\le n$ such that $|S_{ij}|=i+j$ and $S_{ij}\subseteq S_{kl}$ whenever $i\le k$ and $j\le l$. The sets need not be distinct.
Proposed by Ricky Liu. Source: Evan Chen’s third-party solution notes.
Let $m,n$ be coprime positive integers. Initially $m/n$ and $n/m$ are on a blackboard. At each step Evan may choose two numbers $x,y$ on the board and add their arithmetic mean $(x+y)/2$ or their harmonic mean $2xy/(x+y)$. For which pairs $(m,n)$ can he write $1$ after finitely many steps?
Proposed by Yannick Yao. Source: Evan Chen’s third-party solution notes.