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Maximize Sum of Areas in Lattice Grid

Source: 2021 ISL G3

July 12, 2022

Problem Statement

Consider a 100×100100\times 100 square unit lattice <spanclass=′latex−bold′>L</span><span class='latex-bold'>L</span> (hence <spanclass=′latex−bold′>L</span><span class='latex-bold'>L</span> has 1000010000 points). Suppose F\mathcal{F} is a set of polygons such that all vertices of polygons in F\mathcal{F} lie in <spanclass=′latex−bold′>L</span><span class='latex-bold'>L</span> and every point in <spanclass=′latex−bold′>L</span><span class='latex-bold'>L</span> is the vertex of exactly one polygon in F.\mathcal{F}. Find the maximum possible sum of the areas of the polygons in F.\mathcal{F}.
Michael Ren and Ankan Bhattacharya, USA
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