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

(Created page with "Two rational numbers mn and nm are written on a blackboard, where m and n are relatively prime positive integers. At any point, Evan may pi...")
 
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Two rational numbers mn and nm 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 x+y2 or their harmonic mean 2xyx+y on the board as well. Find all pairs (m,n) such that Evan can write 1 on the board in finitely many steps.
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Two rational numbers <math>mn</math> and <math>nm </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>x+y2</math> or their harmonic mean <math>2xyx+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 18:13, 19 April 2019

Two rational numbers $mn$ (Error compiling LaTeX. Unknown error_msg) and $nm$ (Error compiling LaTeX. Unknown error_msg) are written on a blackboard, where $m$ (Error compiling LaTeX. Unknown error_msg) and $n$ (Error compiling LaTeX. Unknown error_msg) are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ (Error compiling LaTeX. Unknown error_msg) and $y$ (Error compiling LaTeX. Unknown error_msg) written on the board and write either their arithmetic mean $x+y2$ (Error compiling LaTeX. Unknown error_msg) or their harmonic mean $2xyx+y$ (Error compiling LaTeX. Unknown error_msg) on the board as well. Find all pairs $(m,n)$ (Error compiling LaTeX. Unknown error_msg) such that Evan can write $1$ on the board in finitely many steps.

Proposed by Yannick Yao