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convergent sum of angle measures

Source: SEEMOUS 2012 P2

June 9, 2021
SummationConvergence

Problem Statement

Let an>0a_n>0, n≥1n\ge1. Consider the right triangles △A0A1A2\triangle A_0A_1A_2, △A0A2A3,…\triangle A_0A_2A_3,\ldots, △A0An−1An,…,\triangle A_0A_{n-1}A_n,\ldots, as in the figure. (More precisely, for every n≥2n\ge2 the hypotenuse A0AnA_0A_n of △A0An−1An\triangle A_0A_{n-1}A_n is a leg of △A0AnAn+1\triangle A_0A_nA_{n+1} with right angle ∠A0AnAn+1\angle A_0A_nA_{n+1}, and the vertices An−1A_{n-1} and An+1A_{n+1} lie on the opposite sides of the straight line A0AnA_0A_n; also, ∣An−1An∣=an|A_{n-1}A_n|=a_n for every n≥1n\ge1.) https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw== Is it possible for the set of points {An∣n≥0}\{A_n\mid n\ge0\} to be unbounded but the series ∑n=2∞m∠An−1A0An\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n to be convergent? Note. A subset BB of the plane is bounded if and only if there is a disk DD such that B⊆DB\subseteq D.
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