Question 3
Coral and Joey 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, the following happens:
• If $n$ is odd, Coral chooses a nonnegative real number $x_{n}$ such that
$$ x_{1}+x_{2}+\cdots+x_{n}\leq\lambda n. $$
• If $n$ is even, Joey chooses a nonnegative real number $x_{n}$ such that
$$ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\leqslant n. $$
All chosen numbers are known to both players. If a player cannot make a move, the game ends and the other player wins.
Determine all values of $ \lambda $ for which Coral has a winning strategy and all those for which Joey has a winning strategy.
• If $n$ is odd, Coral chooses a nonnegative real number $x_{n}$ such that
$$ x_{1}+x_{2}+\cdots+x_{n}\leq\lambda n. $$
• If $n$ is even, Joey chooses a nonnegative real number $x_{n}$ such that
$$ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\leqslant n. $$
All chosen numbers are known to both players. If a player cannot make a move, the game ends and the other player wins.
Determine all values of $ \lambda $ for which Coral has a winning strategy and all those for which Joey has a winning strategy.