IrMO 2004 Problem 10

Suppose $p,q$ are distinct primes and $S$ is a subset of $\{1,2,\dots ,p-1\}$ . Let $N(S)$ denote the number of solutions to the equation $$\sum_{i=1}^{q}x_i\equiv 0\mod p$$ where $x_i\in S$ , $i=1,2,\dots ,q$ . Prove that $N(S)$ is a multiple of $q$ .