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Omni-MATH / Two rational numbers (m)/(n) and (n)/(m) 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 w…

Problem

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problem

Two rational numbers mn\frac{m}{n} and nm\frac{n}{m} are written on a blackboard, where mm and nn are relatively prime positive integers. At any point, Evan may pick two of the numbers xx and yy written on the board and write either their arithmetic mean x+y2\frac{x+y}{2} or their harmonic mean 2xyx+y\frac{2xy}{x+y} on the board as well. Find all pairs (m,n)(m,n) such that Evan can write 11 on the board in finitely many steps.
Plain-text mathematical notation (without MathML)
Two rational numbers (m)/(n) and (n)/(m) 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+y)/(2) or their harmonic mean (2xy)/(x+y) on the board as well. Find all pairs (m,n) such that Evan can write 1 on the board in finitely many steps.
Original LaTeX notation
Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ 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 $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps.

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Source and history

task_id
3f5d9e33-0756-5fa0-a621-5dc5629414be
task_key
test--3f5d9e33-0756-5fa0-a621-5dc5629414be
task_revision_id
5
config / split
/ test

Official source

initial import