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

(Created page with "Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) 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 \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) 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 \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps.
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Two rational numbers <math>\(\tfrac{m}{n}\)</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>\(\tfrac{x+y}{2}\)</math> or their harmonic mean <math>\(\tfrac{2xy}{x+y}\)</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 19:13, 19 April 2019

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

Proposed by Yannick Yao

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