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.

Solution

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Partial Solution:

Consider $P(x)=(x-a)(x-b)(x-c)$ . Let $S_k= a^k+b^k+c^k$ .

According to Newton’s Sum:

$S_{k+3}-(a+b+c)S_{k+2}+(ab+bc+ca)S_{k+1}-(abc)S_k=0$ . So clearly if $a+b+c \vert S_k, S_{k+1},$ then $a+b+c \vert S_{k+3}$ . This proves (b).

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

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2005 Canadian MO Problems/Problem 5

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