2005 Canadian MO Problems/Problem 5

Problem

Let's say that an ordered triple of positive integers $(a,b,c)$ is $n$ -powerful if $a \le b \le c$ , $\gcd(a,b,c) = 1$ , and $a^n + b^n + c^n$ is divisible by $a+b+c$ . For example, $(1,2,2)$ is 5-powerful.

Determine all ordered triples (if any) which are $n$ -powerful for all $n \ge 1$ .
Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.

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2005 Canadian MO (Problems)
Preceded by Problem 4	1 • 2 • 3 • 4 • 5	Followed by Last Question