2019 USAJMO Problems/Problem 6

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Two rational numbers $mn$ (Error compiling LaTeX. Unknown error_msg) and $nm$ (Error compiling LaTeX. Unknown error_msg) are written on a blackboard, where $m$ (Error compiling LaTeX. Unknown error_msg) and $n$ (Error compiling LaTeX. Unknown error_msg) are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ (Error compiling LaTeX. Unknown error_msg) and $y$ (Error compiling LaTeX. Unknown error_msg) written on the board and write either their arithmetic mean $x+y2$ (Error compiling LaTeX. Unknown error_msg) or their harmonic mean $2xyx+y$ (Error compiling LaTeX. Unknown error_msg) on the board as well. Find all pairs $(m,n)$ (Error compiling LaTeX. Unknown error_msg) such that Evan can write $1$ on the board in finitely many steps.

Proposed by Yannick Yao