HMMT 2011 T1

Alice and Barbara play a game on a blackboard. At the start, zero is written on the board. Alice goes first, and the players alternate turns. On her turn, each player replaces $x$ - the number written on the board - with any real number $y$, subject to the constraint that $0<y-x<1$. (a) [10] If the first player to write a number greater than or equal to 2010 wins, determine, with proof, who has the winning strategy. (b) [10] If the first player to write a number greater than or equal to 2010 on her 2011th turn or later wins (if a player writes a number greater than or equal to 2010 on her 2010th turn or earlier, she loses immediately), determine, with proof, who has the winning strategy.