Problem

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2014 3.6

Tags: combinatorics, game, game strategy



$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.