Problem

Source: IMOTC PT1 P3 2018, India

Tags: algebra, Sequence, inequalities



Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$$$b_{n+1}= b_n + \frac{1}{2a_n}$$for all $n\in\mathbb N$. Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.