Question 37
The sequence $\{a_n\}_{n=1}^{\infty}$ of natural numbers satisfies the following relation:
$$
a_{n+2} = \left\lfloor \frac{2a_{n+1}}{a_n} \right\rfloor + \left\lfloor \frac{2a_n}{a_{n+1}} \right\rfloor,
$$
for which by $\lfloor x \rfloor$ we mean the integer part of $x$. Prove that there exists natural number $m$ such that $a_m = 4$ and $a_{m+1} \in \{3, 4\}$.
$$
a_{n+2} = \left\lfloor \frac{2a_{n+1}}{a_n} \right\rfloor + \left\lfloor \frac{2a_n}{a_{n+1}} \right\rfloor,
$$
for which by $\lfloor x \rfloor$ we mean the integer part of $x$. Prove that there exists natural number $m$ such that $a_m = 4$ and $a_{m+1} \in \{3, 4\}$.