US-TST 2009 Problem 8

Fix a prime number $p > 5$ . Let $a,b,c$ be integers no two of which have their difference divisible by $p$ . Let $i,j,k$ be nonnegative integers such that $i + j + k$ is divisible by $p - 1$ . Suppose that for all integers $x$ , the quantity $$(x - a)(x - b)(x - c)[(x - a)^i(x - b)^j(x - c)^k - 1]$$ is divisible by $p$ . Prove that each of $i,j,k$ must be divisible by $p - 1$ .

Kiran Kedlaya and Peter Shor.