A4

Part of 2017 Simon Marais Mathematical Competition

Problems(1)

sum vector is zero vector

Source: Simon Marais 2017 A4

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}{\left\lVert\overrightarrow{PA}_k\right\rVert^5}

is the zero vector. (The notation

\left\lVert\overrightarrow{XY}\right\rVert

represents the length of the vector

\overrightarrow{XY}