IMO 2016 Shortlist C7
Let n ⩾ 2 be an integer. In the plane, there are n segments given in such a way that any two segments have an intersecti...
Category: Combinatorics
Problem
Let n ⩾ 2 be an integer. In the plane, there are n segments given in such a way that any two segments have an intersection point in the interior, and no three segments intersect at a single point. Jeff places a snail at one of the endpoints of each of the segments and claps his hands n−1 times. Each time when he claps his hands, all the snails move along their own segments and stay at the next intersection points until the next clap. Since there are n − 1 intersection points on each segment, all snails will reach the furthest intersection points from their starting points after n − 1 claps. (a) Prove that if n is odd then Jeff can always place the snails so that no two of them ever occupy the same intersection point. (b) Prove that if n is even then there must be a moment when some two snails occupy the same intersection point no matter how Jeff places the snails.