Question 9
Two teams, $A$ and $B$, compete for the territory bounded by a circle.
$A$ has $n$ blue flags and $B$ has $n$ white flags ($n \ge 2$, fixed). They play alternately and $A$ starts the game. Each team, on its turn, places one of its flags at a point of the circle that has not been used in a previous move. Each flag, once placed, cannot be moved.
Once all $2n$ flags have been placed, the territory is divided between the two teams. A point of the territory 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 closest blue flag to a point is at the same distance as the closest white flag to that point, then the point is neutral (it belongs neither to $A$ nor to $B$). A team wins the game if its points cover a larger area than the area 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.
$A$ has $n$ blue flags and $B$ has $n$ white flags ($n \ge 2$, fixed). They play alternately and $A$ starts the game. Each team, on its turn, places one of its flags at a point of the circle that has not been used in a previous move. Each flag, once placed, cannot be moved.
Once all $2n$ flags have been placed, the territory is divided between the two teams. A point of the territory 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 closest blue flag to a point is at the same distance as the closest white flag to that point, then the point is neutral (it belongs neither to $A$ nor to $B$). A team wins the game if its points cover a larger area than the area 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.