NO-ABEL 2017

Let the sequence an be defined by $a_0 = 2, a_1 = 15$ , and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$ . Show that there are infinitely many integers $k$ such that $269 | a_k$ .

Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$ , and the orthocentre $H$ . The point $P$ lies on the ray from $A$ through $O$ . Let $S$ be the mirror image of $P$ through $AC$ , and $T$ the mirror image of $P$ through $AB$ . Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$ .