Let $n$ be an integer that is greater or equal to $12$. Let $P_1, P_2,...,P_n, Q$ be distinct points on the plane. Prove that for some $i$, $\frac{n}{6}-1$ of the distances $P_1P_i, P_2P_i,....P_nP_i$, not including $P_iP_i$, are less than $P_iQ$.
This is where i have got to. Imagine making a pizza with center $Q$.You could rotate the pizza so that none of the points are on the boundary. There are six slices. And so by intermediate pigeon hole principle, there are the roof of $\frac{n}{6}$ in one of the slices. Let $P_i$ be the furthest away from $Q$, now i need to prove that the other points are closer to $P_i$ than $Q$, and this is where I'm stuck...