I found this question on the internet, but I can't find the solution. I believe this question is a pigeon-hole principle type of question. I've tried for specific cases, but I can't prove the statement for all cases of n. Can someone solve prove the statement or at least give a hint of how to do?
" Two regular $n$-gons P and Q ($n\geq 3$) are given in the plane. Prove that the vertices of $P$ that lie inside $Q$ or its boundary are consecutive. (That is, prove that there exist a line separating those vertices of $P$ that lie inside $Q$ or on its boundary from the other vertices of $P$.) "
This isn't true. Consider a square with vertices$$ (\cos \frac{k\pi}{2}, \sin \frac{k\pi}{2}), k=0,1,2,3 $$ and the regular hexagon with vertices at $$ (\cos \frac{k\pi}{3}, \sin \frac{k\pi}{3}), k=0,1,2,3,4,5 $$ The only vertices of one that lie on or inside the other are the vertices they have in common: $(1,0)$ and $(-1,0)$ which are not consecutive vertices of either polygon.