MathDB
Really weird combo geo

Source: St. Petersburg 2023 10.7

August 12, 2023
combinatorics

Problem Statement

Let ℓ1,ℓ2\ell_1, \ell_2 be two non-parallel lines and d1,d2d_1, d_2 be positive reals. The set of points XX, such that dist(X,ℓi)dist(X, \ell_i) is a multiple of did_i is called a <spanclass=′latex−italic′>grid</span><span class='latex-italic'>grid</span>. Let AA be finite set of points, not all collinear. A triangle with vertices in AA is called <spanclass=′latex−italic′>empty</span><span class='latex-italic'>empty</span> if no points from AA lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that AA is the intersection of a grid LL and a convex polygon FF.
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