IMO Shortlist 2001 problem A2


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April 2, 2012
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Let a_0, a_1, a_2, \ldots be an arbitrary infinite sequence of positive numbers. Show that the inequality 1 + a_n > a_{n-1} \sqrt[n]{2} holds for infinitely many positive integers n.
Source: Međunarodna matematička olimpijada, shortlist 2001