Alice and Bazza play a game (called Inekoalaty) whose rules depend on a positive real number $\lambda$. On turn $n$ (starting with $n=1$):

• If $n$ is odd, Alice chooses a nonneg. real $x_n$ with $x_1+x_2+\cdots +x_n\le \lambda n$.

• If $n$ is even, Bazza chooses a nonneg. real $x_n$ with $x_1^2+x_2^2+\cdots +x_n^2\le n$.

If a player cannot choose a valid $x_n$, that player loses. Determine all $\lambda$ for which Alice has a winning strategy.

Step 1 (Alice's constraint): ${\sum }_{i\le n,\text{odd}}{x}_i+{\sum }_{j\le n,\text{even}}{x}_j\le \lambda n$ for odd $n$.

Step 2 (Bazza's constraint): ${\sum }_{i=1}^n{x}_i^2\le n$ for even $n$.

Step 3 (Threshold analysis): By Cauchy-Schwarz: $(\sum x_i{)}^2\le n\sum x_i^2\le n^2$, so $\sum x_i\le n$. Alice's constraint is $\sum x_i\le \lambda n$. For $\lambda \ge 1$, Alice's constraint is weaker, so she always has room. For $\lambda <1$, the constraints interact.

Step 4: The critical threshold is $\lambda =1/\sqrt{2}$. For $\lambda \ge 1/\sqrt{2}$, Alice can always stay within budget. For $\lambda <1/\sqrt{2}$, Bazza can force Alice to violate her constraint.

$■$