2019 USAMO 第 5 题

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.

Proposed by Yannick Yao

黑板上写着两个有理数$\frac{m}{n}$和$\frac{n}{m}$，其中$m$和$n$是互素正整数。在任何时候，埃文都可以选择写在黑板上的数字 $x$ 和 $y$ 中的两个，并将它们的算术平均值 $\frac{x+y}{2}$ 或它们的调和平均值 $\frac{2xy}{x+y}$ 写在黑板上。找到所有 $(m,n)$ 对，使得 Evan 可以通过有限步在黑板上写下 $1$。

由 Yannick Yao 提议