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.
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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
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==Solution==
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We claim that all odd <math>m, n</math> work if <math>m+n</math> is a positive power of 2.
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Proof:
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We first prove that <math>m+n=2^k</math> works. By weighted averages we have that <math>\frac{n(\frac{m}{n})+(2^k-n)\frac{n}{m}}{2^k}=\frac{m+n}{2^k}=1</math> can be written, so the solution set does indeed work. We will now prove these are the only solutions.
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Assume that <math>m+n\ne 2^k</math>, so then <math>m+n\equiv 0\pmod{p}</math> for some odd prime <math>p</math>. Then <math>m\equiv -n\pmod{p}</math>, so <math>\frac{m}{n}\equiv \frac{n}{m}\equiv -1\pmod{p}</math>. We see that the arithmetic mean is <math>\frac{-1+(-1)}{2}\equiv -1\pmod{p}</math> and the harmonic mean is <math>\frac{2(-1)(-1)}{-1+(-1)}\equiv -1\pmod{p}</math>, so if 1 can be written then <math>1\equiv -1\pmod{p}</math> and <math>2\equiv 0\pmod{p}</math> which is obviously impossible, and we are done.
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-Stormersyle 
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{{MAA Notice}}
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==See also==
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{{USAJMO newbox|year=2019|num-b=5|aftertext=|after=Last Problem}}

Latest revision as of 23:57, 19 April 2019

Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ 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

Solution

We claim that all odd $m, n$ work if $m+n$ is a positive power of 2.

Proof: We first prove that $m+n=2^k$ works. By weighted averages we have that $\frac{n(\frac{m}{n})+(2^k-n)\frac{n}{m}}{2^k}=\frac{m+n}{2^k}=1$ can be written, so the solution set does indeed work. We will now prove these are the only solutions.

Assume that $m+n\ne 2^k$, so then $m+n\equiv 0\pmod{p}$ for some odd prime $p$. Then $m\equiv -n\pmod{p}$, so $\frac{m}{n}\equiv \frac{n}{m}\equiv -1\pmod{p}$. We see that the arithmetic mean is $\frac{-1+(-1)}{2}\equiv -1\pmod{p}$ and the harmonic mean is $\frac{2(-1)(-1)}{-1+(-1)}\equiv -1\pmod{p}$, so if 1 can be written then $1\equiv -1\pmod{p}$ and $2\equiv 0\pmod{p}$ which is obviously impossible, and we are done.

-Stormersyle

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See also

2019 USAJMO (ProblemsResources)
Preceded by
Problem 5
Last Problem
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All USAJMO Problems and Solutions
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