Let n be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter 6n, and 60∘ rotational symmetry (that is, there is a point O such that a 60∘ rotation about O maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its 3n2+3n+1 citizens at 3n2+3n+1 points in the kingdom so that every two citizens have a distance of at least 1 for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.)Proposed by Ankan Bhattacharya, USA RMM 2021geometrycombinatoricscombinatorial geometryRMM