2019 USAJMO Problems/Problem 6

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Two rational numbers $\(\tfrac{m}{n}\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) and $\(\tfrac{n}{m}\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) are written on a blackboard, where $\(m\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) and $\(n\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) are relatively prime positive integers. At any point, Evan may pick two of the numbers $\(x\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) and $\(y\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) written on the board and write either their arithmetic mean $\(\tfrac{x+y}{2}\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) or their harmonic mean $\(\tfrac{2xy}{x+y}\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) on the board as well. Find all pairs $\((m,n)\)$ (Error compiling LaTeX. ! LaTeX Error: Bad math environment delimiter.) such that Evan can write $1$ on the board in finitely many steps.

Proposed by Yannick Yao

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