IMO Shortlist 1991 problem 12


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2. travnja 2012.
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Given any integer n \geq 2, assume that the integers a_1, a_2, \ldots, a_n are not divisible by n and, moreover, that n does not divide \sum^n_{i=1} a_i. Prove that there exist at least n different sequences (e_1, e_2, \ldots, e_n) consisting of zeros or ones such \sum^n_{i=1} e_i \cdot a_i is divisible by n.
Izvor: Međunarodna matematička olimpijada, shortlist 1991