Problem

Source: Iran TST 2012-Third exam-1st day-P1

Tags: geometry, geometric transformation, reflection, perimeter, rotation, induction, geometry proposed



Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon. Proposed by Hesam Rajabzade