International Mathematical Olympiad 1986 Problem 2

A triangle $A_{1}A_{2}A_{3}$ and a point $P_{0}$ are given in the plane. We define $A_{s} = A_{s-3}$ for all $s \geq 4$. We construct a set of points $P_{1}, P_{2}, P_{3}, \ldots$, such that $P_{k+1}$ is the image of $P_{k}$ under a rotation with center $A_{k+1}$ through angle $120^{\circ}$ clockwise (for $k = 0, 1, 2, \ldots$). Prove that if $P_{1986} = P_{0}$, then the triangle $A_{1}A_{2}A_{3}$ is equilateral.