If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$ 
Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$.
  Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$.

(problem composed by Laurentiu Panaitopol)
So far no idea.

It seems that there's a partial solution. 
Suppose that $\mathrm{gcd}(a,a+b+c)=\mathrm{gcd}(b,a+b+c)=\mathrm{gcd}(c,a+b+c)=1$. 
Then for $n=k\cdot \phi(a+b+c)+1 \, (k=1,2, \ldots )$, where $\phi$ is Euler's function, we have:
$$
(a^n+b^n+c^n)-(a^2+b^2+c^2)=a^2 (a^{n-1}-1) + b^2 (b^{n-1}-1) + 
c^2 (c^{n-1}-1),
$$ 
where all round brackets are divisible by $a+b+c$ according to Euler theorem. 
Therefore $(a+b+c) \mid (a^n+b^n+c^n)$ for all these $n$. 
