IMO 2018 Shortlist C7
Consider 2018 pairwise rossing ir les no three of whi h are on urrent. These ir les subdivide the plane into regions bou...
Category: Combinatorics
Problem
Consider 2018 pairwise rossing ir les no three of whi h are on urrent. These ir les subdivide the plane into regions bounded by ir ular edges that meet at verti es. Noti e that there are an even number of verti es on ea h ir le. Given the ir le, alternately olour the verti es on that ir le red and blue. In doing so for ea h ir le, every vertex is oloured twi e on e for ea h of the two ir les that ross at that point. If the two olourings agree at a vertex, then it is assigned that olour; otherwise, it be omes yellow. Show that, if some ir le ontains at least 2061 yellow points, then the verti es of some region are all yellow. (India) 6 Cluj-Napo a Romania, 314 July 2018