Blastoor wrote: Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$. Prove that the sequence does not contain numbers $5$ and $11$. Note that $a_1=2,a_2=3$. Thus $2\nmid a_1a_2\cdots a_n+1,3\nmid a_1a_2\cdots a_n+1$. Thus $a_k,k\geq2$ is odd. If $a_{k+1}=5$, we have $a_1a_2\cdots a_n+1=5^m$ for some $m\in\mathbb{N}$ and $2\equiv a_1a_2\cdots a_n=5^m-1\equiv0\pmod{4}$. Thus the sequnce does not contain 5. The rest is Germany 2018.