Difference between revisions of "2019 USAJMO Problems/Problem 6"
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− | Two rational numbers <math>frac{m}{n}</math> and <math> | + | Two rational numbers <math>\frac{m}{n}</math> and <math>\frac{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 18:14, 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