2007 OIM Problems/Problem 3

Problem

Two teams, $A$ and $B$ , dispute the territory limited by a circle. $A$ has $n$ blue flags and $B$ has $n$ white flags ( $n \ge 2$ , fixed). They play alternately and the game begins. Each team, in turn, places one of its flags at a point on the circumference that has not been used in a previous play. Once a flag is placed you can't change its location. Once the $2n$ flags have been placed, the territory is divided between the two teams. A territory point belongs to team $A$ if the flag closest to it is blue, and to team $B$ if the flag closest to it is white. If the blue flag closest to a point is the same distance as the nearest white flag to that point, then the point is neutral (it is neither $A$ nor $B$ ). $A$ team wins the game if its points cover an area greater than the area covered by the other team's points. There is a tie if both areas covered are equal. Show that, for all $n$ , team $B$ has a strategy to win the game.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2007 OIM Problems/Problem 3

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