Baltic Way 2019 · Problem 15

Let $n\ge4$, and consider a (not necessarily convex) polygon $P_1P_2\ldots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\ldots,P_n$ that lies closest to it. The polygon is then said to be hostile if $Q_k\ne P_{k+1}$ for all $k$ (where $P_0=P_n$, $P_{n+1}=P_1$).

(a) Prove that no hostile polygon is convex.

(b) Find all $n\ge4$ for which there exists a hostile $n$-gon.