ISL 1997 Problem 6

(a) Let $n$ be a positive integer. Prove that there exist distinct positive integers $x, y, z$ such that

$$x^{n-1} + y^n = z^{n+1}.$$

(b) Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or to $b.$ Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers $x, y, z$ such that

$$x^a + y^b = z^c.$$