International Mathematical Olympiad 2001 Problem 4

Let $n$ be an odd integer greater than 1, and let $k_1, k_2, \ldots, k_n$ be given integers. For each of the $n!$ permutations $a = (a_1, a_2, \ldots, a_n)$ of $1, 2, \ldots, n$, let

$$S(a) = \sum_{i=1}^{n} k_i a_i.$$

Prove that there are two permutations $b$ and $c, b \neq c$, such that $n!$ is a divisor of $S(b) - S(c)$.