Difference between revisions of "2007 OIM Problems/Problem 3"
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| − | Two teams, <math> | + | Two teams, <math>A</math> and <math>B</math>, dispute the territory limited by a circle. <math>A</math> has <math>n</math> blue flags and <math>B</math> has <math>n</math> white flags (<math>n \ge 2</math>, 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 <math>2n</math> flags have been placed, the territory is divided between the two teams. A territory point belongs to team <math>A</math> if the flag closest to it is blue, and to team <math>B</math> 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 <math>A</math> nor <math>B</math>). <math>A</math> 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 <math>n</math>, team <math>B</math> has a strategy to win the game. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 15:46, 14 December 2023
Problem
Two teams, 
and 
, dispute the territory limited by a circle. has 
blue flags and has white flags (
, 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 
flags have been placed, the territory is divided between the two teams. A territory point belongs to team if the flag closest to it is blue, and to team 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 nor ). 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 , team has a strategy to win the game.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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