Difference between revisions of "2019 USAJMO Problems/Problem 6"
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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. | + | 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 18:13, 19 April 2019
Two rational numbers $\(\tfrac{m}{n}\)$ (Error compiling LaTeX. Unknown error_msg) and $\(\tfrac{n}{m}\)$ (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 $\(\tfrac{x+y}{2}\)$ (Error compiling LaTeX. Unknown error_msg) or their harmonic mean $\(\tfrac{2xy}{x+y}\)$ (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 
on the board in finitely many steps.
Proposed by Yannick Yao
