SCHWEITZER 2010 Problem 1

Let $p$ be prime. Denote by $N (p)$ the number of integers $x$ for which $1 \leq x \leq p$ and $$x ^ {x} \equiv 1 (\bmod p)$$ Prove that there exist numbers $c <1/2$ and $p_ {0}> 0$ such that $$N (p) \leq p ^ {c}$$ if $p \ge p_ {0}$ .