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Show that there exists a permutation

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September 9, 2010
trigonometrynumber theoryequationTrigonometric EquationsTrigonometric IdentitiesIMO Shortlist

Problem Statement

Let nn be a positive integer having at least two different prime factors. Show that there exists a permutation a1,a2,…,ana_1, a_2, \dots , a_n of the integers 1,2,…,n1, 2, \dots , n such that ∑k=1nk⋅cos⁡2πakn=0.\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.
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