markdown/2025/p5/problem.md

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 $n^{\text{th}}$ 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

$$x_1+x_2+\cdots +x_n\le \lambda n\mathrm{.}x\_1\; +\; x\_2\; +\; \backslash cdots\; +\; x\_n\; \backslash leq\; \backslash lambda\; n.$$

• If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that

$$x_1^2+x_2^2+\cdots +x_n^2\le n\mathrm{.}x\_1^2\; +\; x\_2^2\; +\; \backslash cdots\; +\; x\_n^2\; \backslash leq\; n.$$

If a player cannot choose a suitable number $x_n$, the game ends and the other player wins. If the game goes on 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.