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 July 21st, 2009, 02:37 AM #1 Joined: Apr 2007 Posts: 2,141 Thanks: 0 The 8th Korean Mathematical Olympiad Since I'm Korean, I will post this interesting problem by KMO. 2. For a given positive integer $m$, find all pairs $(n, x, y)$ of positive integers such that $m$, $n$ are relatively prime and $(x^2+y^2)^m= (xy)^n$, where $n$, $x$, $y$ can be represented by functions of $m$.
 July 22nd, 2009, 04:08 PM #2 Joined: Apr 2007 Posts: 2,141 Thanks: 0 Re: The 8th Korean Mathematical Olympiad Hint: this could require basic number theory knowledge.
July 22nd, 2009, 09:07 PM   #3
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Re: The 8th Korean Mathematical Olympiad

Quote:
 Originally Posted by johnny where $n$, $x$, $y$ can be represented by functions of $m$.
What does this mean?

July 22nd, 2009, 09:47 PM   #4

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Re: The 8th Korean Mathematical Olympiad

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by johnny where $n$, $x$, $y$ can be represented by functions of $m$.
What does this mean?
For example, m(x) = n+x+y?

July 23rd, 2009, 08:18 AM   #5
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Re: The 8th Korean Mathematical Olympiad

Quote:
 Originally Posted by johnny For example, m(x) = n+x+y?
So "where n, x, y can be represented by functions of m" means "m is a function of n, x, and y"?

July 23rd, 2009, 02:12 PM   #6

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Re: The 8th Korean Mathematical Olympiad

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by johnny For example, m(x) = n+x+y?
So "where n, x, y can be represented by functions of m" means "m is a function of n, x, and y"?
I think so, but I'm 75% sure about it.