imo-5-gpt

Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:

• If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that $$x_1+x_2+\cdots+x_n\le\lambda n.$$
• If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that $$x_1^2+x_2^2+\cdots+x_n^2\le n.$$

If a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.