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IMO ShortList 2001, algebra problem 2

Source: IMO ShortList 2001, algebra problem 2

September 30, 2004
inequalitiescalculusIMO Shortlist

Problem Statement

Let a0,a1,a2,…a_0, a_1, a_2, \ldots be an arbitrary infinite sequence of positive numbers. Show that the inequality 1+an>an−12n1 + a_n > a_{n-1} \sqrt[n]{2} holds for infinitely many positive integers nn.
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