Question 97
Let $n \ge 4$, and consider a non-intersecting $n$-gon $P_1P_2...P_n$ in the plane. Suppose that, to each $P_k$, there is a unique other vertex $Q_k$ among $P_1, ..., P_n$ that lies closest to it. The polygon is said to be *hostile* if $Q_k \ne P_{k \pm 1}$ for all $k$ (counting cyclically).
a. Prove that there exist no convex hostile polygons.
b. Determine all $n$ for which there exists a concave hostile $n$-gon.
a. Prove that there exist no convex hostile polygons.
b. Determine all $n$ for which there exists a concave hostile $n$-gon.