A geometry task for problem solvers (involving circle, points & sections)

Question

Any help regarding how I should tackle this problem is appreciated.

We have a circle with the radius = 1. We also have dots, lets call them $P_1, P_2,.....,P_n $. So we have n numbers of dots, where $n \ge 1$. Show that you can choose a spot (lets call this dot Q) on the circle's edge, so that $QP_1+QP_2+...+QP_n \ge n$.

Answer 1

Without loss of generality, assume the circle is the standard unit circle.

Let $V = P_1 + \cdots + P_n$.

If $V=0$, choose any point $Q$ on the circle, else choose $Q = -{\large{\frac{V}{|V|}}}$.

\begin{align*} \text{Then}\;\; &QP_1 + \cdots +QP_n\\[4pt] =\;&|Q-P_1| + \cdots + |Q-P_n|\\[4pt] \ge\;&|nQ - V|\\[4pt] \ge\;&|nQ - 0|\\[4pt] =\;&n\\[4pt] \end{align*}