APMO 2020 P2
Posted: Thu Dec 03, 2020 9:21 pm
Show that $r = 2$ is the largest real number $r$ which satisfies the following condition:
If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.
If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities
\[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\]for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.