# Difference between revisions of "2019 USAJMO Problems/Problem 6"

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

Proposed by Yannick Yao | Proposed by Yannick Yao |

## Revision as of 19:14, 19 April 2019

Two rational numbers and are written on a blackboard, where and are relatively prime positive integers. At any point, Evan may pick two of the numbers and written on the board and write either their arithmetic mean or their harmonic mean on the board as well. Find all pairs such that Evan can write on the board in finitely many steps.

Proposed by Yannick Yao