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Non-adjacent polygonal faces

Source: IMAR Test 2023 P2

December 16, 2023
combinatoricscombinatorial geometry

Problem Statement

Consider n⩾6n\geqslant 6 coplanar lines, no two parallel and no three concurrent. These lines split the plane into unbounded polygonal regions and polygons with pairwise disjoint interiors. Two polygons are non-adjacent if they do not share a side. Show that there are at least (n−2)(n−3)/12(n-2)(n-3)/12 pairwise non-adjacent polygons with the same number of sides each.
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