SEEMOUS 2012 Problem 2

Let $a_n>0$ , $n\ge1$ . Consider the right triangles $\triangle A_0A_1A_2$ , $\triangle A_0A_2A_3,\ldots$ , $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$ , and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$ ; also, $|A_{n-1}A_n|=a_n$ for every $n\ge1$ .) Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? Note. A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$ .