IMC 2001 · Problem 4

Let $k$ be a positive integer. Let $p(x)$ be a polynomial of degree $n$ each of whose coefficients is $-1$, $1$ or $0$, and which is divisible by $(x-1)^k$. Let $q$ be a prime such that $\dfrac{q}{\ln q} < \dfrac{k}{\ln(n+1)}$. Prove that the complex $q$th roots of unity are roots of the polynomial $p(x)$.