Question 156
Fix a prime number $p \gt 5$. 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 $p-1$.
$$
(x-a)(x-b)(x-c)[(x-a)^i(x-b)^j(x-c)^k - 1]
$$
is divisible by $p$. Prove that each of $i, j, k$ must be divisible by $p-1$.