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inequality in sequences

Source: Mongolian MO 2002 Grade 10 P5

April 9, 2021
inequalitiesSequence

Problem Statement

Let a0,a1,…a_0,a_1,\ldots be an infinite sequence of positive numbers. Prove that the inequality 1+an>2nan−11+a_n>\sqrt[n]2a_{n-1} holds for infinitely many positive integers nn.
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