2025 IMO 第 5 题

Alice and Bazza are playing the inekolaty 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

$$x_{1} + x_{2} + \dots +x_{n}\leq \lambda n.$$

- If $n$ is even, Bazza chooses a nonnegative real number $x_{n}$ such that

$$x_{1}^{2} + x_{2}^{2} + \dots +x_{n}^{2}\leq n.$$

If a player cannot choose a suitable $x_{n}$ , 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.

Alice 和 Bazza 正在玩 inekolaty 游戏，这是一种两人游戏，其规则取决于两个玩家都知道的正实数 $\lambda$。在游戏的第 $n$ 回合(从 $n = 1$ 开始)，会发生以下情况：

- 如果 $n$ 是奇数，Alice 选择一个非负实数 $x_{n}$ 使得

$$x_{1} + x_{2} + \dots +x_{n}\leq \lambda n.$$

- 如果 $n$ 是偶数，Bazza 选择一个非负实数 $x_{n}$ 使得

$$x_{1}^{2} + x_{2}^{2} + \dots +x_{n}^{2}\leq n.$$

如果一名玩家无法选择合适的 $x_{n}$ ，则游戏结束，另一名玩家获胜。如果游戏永远持续下去，则双方都不会获胜。所有选择的号码都是双方玩家都知道的。

确定 Alice 有获胜策略的 $\lambda$ 的所有值以及 Bazza 有获胜策略的所有值。