Question 15
We have in the plane a point $P_{0}$ and a triangle $A_{1} A_{2} A_{3}$. We define $A_{s}=A_{s-3}$ for all $s \geq 4$. We build a succession of $P_{1}, P_{2}, P_{3}, \ldots$ points, so that $P_{k+1}$ is the image of $P_{k}$ by the center rotation $A_{k+1}$ and angle $120^{\circ}$ clockwise, for $k=0,1,2 \ldots$ to show that if $P_{1986}=P_{0}$, then triangle $A_{1} A_{2} A_{3}$ is equilateral.