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recursive sequence defined by floor function

Source: Iran second round- 2013- P6

May 4, 2013
functioninequalitiesalgebra proposedalgebraRecurrence

Problem Statement

Let {an}n=1∞\{a_n\}_{n=1}^{\infty} be a sequence of positive integers for which an+2=[2anan+1]+[2an+1an]. a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. Prove that there exists a positive integer mm such that am=4a_m=4 and am+1∈{3,4}a_{m+1} \in\{3,4\}.
Note. [x][x] is the greatest integer not exceeding xx.
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