USAJMO 2019 Question 1

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.