SIMON-MARAIS 2017 A4

Let $A_1, A_2, \ldots, A_{2017}$ be the vertices of a regular polygon with 2017 sides.

Prove that there exists a point $P$ in the plane of the polygon such that the vector

$$\sum_{k=1}^{2017} k \frac{\overrightarrow{PA_k}}{\|\overrightarrow{PA_k}\|^5}$$

is the zero vector.

(The notation $\|\overrightarrow{XY}\|$ represents the length of the vector $\overrightarrow{XY}$.)