CN-TST 2018 Problem 22

Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$ , and there exists positive integer $x$ such that $x^k\equiv a \pmod p$ . Suppose that $3\le |S| \le p-2$ . Prove that the elements of $S$ , when arranged in increasing order, does not form an arithmetic progression.