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A sequence

Source: Baltic Way 2004 problem 1

November 20, 2004
inequalitiesalgebra solvedalgebra

Problem Statement

Given a sequence a1,a2,…a_1,a_2,\ldots of non-negative real numbers satisfying the conditions:
1. an+a2n≥3na_n + a_{2n} \geq 3n; 2. an+1+n≤2an(n+1)a_{n+1}+n \leq 2\sqrt{a_n \left(n+1\right)}
for all n∈Nn\in\mathbb N (where N={1,2,3,...}\mathbb N=\left\{1,2,3,...\right\}).
(1) Prove that the inequality an≥na_n \geq n holds for every n∈Nn \in \mathbb N. (2) Give an example of such a sequence.
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