MathDB

Two rational numbers

\tfrac{m}{n}

and

\tfrac{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

\tfrac{x+y}{2}

or their harmonic mean

\tfrac{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.
Proposed by Yannick Yao

Problem Statement