69th Belarusian Mathematical Olympiad

First five terms of the sequence are $a_1=1$, $a_2=1$, $a_3=2$, $a_4=3$ and $a_5=3.5$. $$\text{Note that }{a}_{n+2}^2=a_{n+1}^2+\frac{2a_{n+1}}{a_n}+\frac{1}{a_n^2}=a_{n+1}^2+2+\frac{2}{a_{n-1}{a}_n}+\frac{1}{a_n^2}>a_{n+1}^2+2.$$ Therefore, $a_{180}^2>a_5^2+350=362.25>19^2$, whence $a_{180}>19$.