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. geometrycombinatorics proposedcombinatorics