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min network in equiangular hexagon (V Soros Olympiad 1998-99 Round 1 10.9)

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May 25, 2024
geometrygeometric inequalityhexagon

Problem Statement

Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length aa, and the remaining three have length bb (aba \le b). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.