Problem

Source: 2019 USAMO Problem 5, 2019 USAJMO Problem 6

Tags: number theory, relatively prime, USA(J)MO, USAMO, USAJMO, Hi



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