IR-TST 2012 Problem 13

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