## The 8th Korean Mathematical Olympiad

IMO, IMC, USAMO, TST, Putnam, post your own math challenges to the group.

### The 8th Korean Mathematical Olympiad

Since I'm Korean, I will post this interesting problem by KMO.

2. For a given positive integer [latex]m[/latex], find all pairs [latex](n, x, y)[/latex] of positive integers such that [latex]m[/latex], [latex]n[/latex] are relatively prime and [latex](x^2+y^2)^m = (xy)^n[/latex], where [latex]n[/latex], [latex]x[/latex], [latex]y[/latex] can be represented by functions of [latex]m[/latex].
johnny
Retired Moderator

Posts: 2141
Joined: Tue Apr 24, 2007 9:43 am

### Re: The 8th Korean Mathematical Olympiad

Hint: this could require basic number theory knowledge.
johnny
Retired Moderator

Posts: 2141
Joined: Tue Apr 24, 2007 9:43 am

### Re: The 8th Korean Mathematical Olympiad

johnny wrote:where [latex]n[/latex], [latex]x[/latex], [latex]y[/latex] can be represented by functions of [latex]m[/latex].

What does this mean?
Pari/GP: this is the program I probably mentioned in my post. Windows users can get it at http://pari.math.u-bordeaux.fr/pub/pari ... -2-6-1.exe

CRGreathouse
Global Moderator

Posts: 12061
Joined: Sat Nov 25, 2006 9:52 am
Location: UTC -5

### Re: The 8th Korean Mathematical Olympiad

CRGreathouse wrote:
johnny wrote:where [latex]n[/latex], [latex]x[/latex], [latex]y[/latex] can be represented by functions of [latex]m[/latex].

What does this mean?

For example, m(x) = n+x+y?
johnny
Retired Moderator

Posts: 2141
Joined: Tue Apr 24, 2007 9:43 am

### Re: The 8th Korean Mathematical Olympiad

johnny wrote: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"?
Pari/GP: this is the program I probably mentioned in my post. Windows users can get it at http://pari.math.u-bordeaux.fr/pub/pari ... -2-6-1.exe

CRGreathouse
Global Moderator

Posts: 12061
Joined: Sat Nov 25, 2006 9:52 am
Location: UTC -5

### Re: The 8th Korean Mathematical Olympiad

CRGreathouse wrote:
johnny wrote: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.
johnny
Retired Moderator

Posts: 2141
Joined: Tue Apr 24, 2007 9:43 am