IMO Shortlist 2009 problem N1


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2. travnja 2012.
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Let n be a positive integer and let a_1, a_2, a_3, ..., a_k (k \geqslant 2) be distinct integers in the set \left\{1,\,2,\,\ldots,\,n\right\} such that n divides a_i \left(a_{i + 1} - 1\right) for i = 1,\,2,\,\ldots,\,k - 1. Prove that n does not divide a_k \left(a_1 - 1\right).

Proposed by Ross Atkins, Australia
Izvor: Međunarodna matematička olimpijada, shortlist 2009