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
