Question 22
(SWE 4) Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1,a_2,\ldots,a_n$ of the integers $1,2,\ldots,n$ such that
$$\sum_{k=1}^n k\cdot\cos\frac{2\pi a_k}{n}=0.$$
$$\sum_{k=1}^n k\cdot\cos\frac{2\pi a_k}{n}=0.$$