USAJMO 2019 Question 6

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.