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sum vector is zero vector

Source: Simon Marais 2017 A4

April 29, 2021
vector

Problem Statement

Let A1,A2,…,A2017A_1,A_2,\ldots,A_{2017} be the vertices of a regular polygon with 20172017 sides.Prove that there exists a point PP in the plane of the polygon such that the vector ∑k=12017kPA→k∥PA→k∥5\sum_{k=1}^{2017}k\frac{\overrightarrow{PA}_k}{\left\lVert\overrightarrow{PA}_k\right\rVert^5} is the zero vector. (The notation ∥XY→∥\left\lVert\overrightarrow{XY}\right\rVert represents the length of the vector XY→\overrightarrow{XY}.)
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