SHARYGIN 2023 Problem 14

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$ , and is symmetric with respect to $O$ . Prove that the winding number of $\mathcal{L}$ around $O$ is odd.

The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$ : $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$