Question 4
Let $n \gt 1$ be an odd integer and let $c_1$, $c_2$, …, $c_n$ be integers. For each permutation $a = (a_1, a_2, \dots, a_n)$ of $\{1,2,\dots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist two permutations $a \neq b$ of $\{1,2,\dots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.