Question 5
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 $x_n$, 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.
• 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 $x_n$, 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.