BG-MO 1989 Problem 6

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions:

(i) $6p$ does not divide $q-1$ ; (ii) $q$ divides $x^2+xy+y^2$ ; (iii) $q$ does not divide $x+y-z$ .

Prove that $x^p+y^p\ne z^p$ .