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

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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.
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Two rational numbers <math>frac{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>\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 19:14, 19 April 2019

Two rational numbers $frac{m}{n}$ 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 $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps.

Proposed by Yannick Yao

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