CMO 2018 Problem 1

Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying $$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$ are both integers that are relatively prime to $p$ . If $A_n$ is finite, let $f(n)$ denote $|A_n|$ .

a) Prove that $A_n$ is finite if and only if $n \not = 2$ .

b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that $$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$