2019 USAJMO Problems/Problem 6

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

Proposed by Yannick Yao