OIM 2007 Problem 3

Two teams, $A$ and $B$ , fight for a territory limited by a circumference.

$A$ has $n$ blue flags and $B$ has $n$ white flags ( $n\geq 2$ , fixed). They play alternatively and $A$ 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 $2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $A$ if the closest flag to it is blue, and it belongs to $B$ 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 $A$ nor from $B$ ). 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 $n$ , team $B$ has a winning strategy.