Question 169
Let $n$ be an odd integer greater than $1$ and let $c_1, c_2, \dots, c_n$ 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 permutations $b$ and $c$, $b \neq c$, such that $n!$ divides $S(b) - S(c)$.