Find the maximun $n\in N^{+}$, such $x^n+y^n+z^n$ is divisible by $x+y+z$ A prime number $p$ and integers $x,y,z$ with $0<x<y<z<p$ are given. Show that if the numbers $x^3,y^3,z^3$ give the same remainder when divided by $p$ , Find the maximun $n\in N^{+}$, such $x^n+y^n+z^n$ is divisible by $x+y+z$
when $n=2$ case be see Poland 2003

I think you have to edit your post.
Are you sure you don’t ask about the maximum $n$ such that all of the $x^m+y^m+z^m$ with $m> n$  are not divisible by $x+y+z$?
In your hyperlink,  Poland 2003, it is not question of maximum n but just for $n=2$.
See at the following example: $0<3<5<6<7$ and $3^3, 5^3, 6^3$ have the same residue, $-1$ modulo $7$.
In fact, $3^2+5^2+6^2= 5(3+5+6)$, this is true. 
But you have also $3^4+5^4+6^4=143(3+5+6)$ so $n=2$ is not a maximum as required. And it is not proved in any way that $n=4$ is. 
