Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be an odd integer greater than 1 and let
![c_1, c_2, \ldots, c_n](/media/m/7/a/f/7afa08c403de0a0e4c2c68e3910089c9.png)
be integers. For each permutation
![a = (a_1, a_2, \ldots, a_n)](/media/m/4/5/c/45c463244a8ee5dda3ef9ffa2c86b732.png)
of
![\{1,2,\ldots,n\}](/media/m/9/6/a/96a5b5240166685da05abd76f5be8328.png)
, define
![S(a) = \sum_{i=1}^n c_i a_i](/media/m/b/1/d/b1d97455fcfe1b2a71a14a57aa1a93e0.png)
. Prove that there exist permutations
![a \neq b](/media/m/1/0/9/1098169ff6b22300ecf0b743facabd41.png)
of
![\{1,2,\ldots,n\}](/media/m/9/6/a/96a5b5240166685da05abd76f5be8328.png)
such that
![n!](/media/m/5/e/9/5e9bb819f1bfbf465700f6bc8831a1c7.png)
is a divisor of
![S(a)-S(b)](/media/m/9/7/2/97269b1167800c59f63630f5de0ee769.png)
.
%V0
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.