MathDB

Let

n

be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter

6n

, and

60^\circ

rotational symmetry (that is, there is a point

O

such that a

60^\circ

rotation about

O

maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its

3n^2+3n+1

citizens at

3n^2+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

Problem Statement