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Flag strategy at a circle

Source:

September 12, 2007
geometryalgorithminductioncombinatorics proposedcombinatorics

Problem Statement

Two teams, A A and B B, fight for a territory limited by a circumference. A A has n n blue flags and B B has n n white flags (n≥2 n\geq 2, fixed). They play alternatively and A A begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved. Once all 2n 2n flags have been placed, territory is divided between the two teams. A point of the territory belongs to A A if the closest flag to it is blue, and it belongs to B B if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from A A nor from B B). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas. Prove that, for every n n, team B B has a winning strategy.
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