ILL 1974 Problem 10

A regular octagon $P$ is given whose incircle $k$ has diameter $1$ . About $k$ is circumscribed a regular $16$ -gon, which is also inscribed in $P$ , cutting from $P$ eight isosceles triangles. To the octagon $P$ , three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$ -gon so obtained is said to be $P'$ . Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$ , one of the $11$ -gons $P'$ contains all of $M$ .