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Romania District Olympiad 2010

Source: Grade X

March 13, 2010
searchfunctionalgebra proposedalgebra

Problem Statement

Consider the sequence a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1, where z∈C∗ z\in \mathbb{C}^* is given. i) Prove that if a1>2 a_1>2, then: a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^* ii) Prove that if there is a k∈N∗ k\in \mathbb{N}^* such that ak≤2 a_k\le 2, then a1≤2 a_1\le 2.
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