IMO Shortlist 1983 problem 22
Dodao/la:
arhiva2. travnja 2012. Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a positive integer having at least two different prime factors. Show that there exists a permutation
![a_1, a_2, \dots , a_n](/media/m/6/a/b/6abf1d6c29ff9c7cfa81513d24f7324e.png)
of the integers
![1, 2, \dots , n](/media/m/a/9/c/a9c1fa4f9842bafaf5cd14a9a1352c50.png)
such that
%V0
Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that
$$\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1983