Difference between revisions of "2025 IMO Problems/Problem 5"
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Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number <math>\lambda</math> which is known to both players. On the <math>n</math>th turn of the game (starting with <math>n=1</math>) the following happens: | Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number <math>\lambda</math> which is known to both players. On the <math>n</math>th turn of the game (starting with <math>n=1</math>) the following happens: | ||
| + | |||
If <math>n</math> is odd, Alice chooses a nonnegative real number <math>x_n</math> such that | If <math>n</math> is odd, Alice chooses a nonnegative real number <math>x_n</math> such that | ||
| − | + | ||
| − | x_1 + x_2 + \cdots + x_n \le \lambda n | + | <cmath>x_1 + x_2 + \cdots + x_n \le \lambda n</cmath> |
| − | + | ||
If <math>n</math> is even, Bazza chooses a nonnegative real number <math>x_n</math> such that | If <math>n</math> is even, Bazza chooses a nonnegative real number <math>x_n</math> such that | ||
| − | + | ||
| − | x_1^2 + x_2^2 + \cdots + x_n^2 \le n | + | <cmath>x_1^2 + x_2^2 + \cdots + x_n^2 \le n</cmath> |
| − | + | ||
If a player cannot choose a suitable <math>x_n</math>, 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. | If a player cannot choose a suitable <math>x_n</math>, 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. | ||
Revision as of 06:57, 16 July 2025
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number 
which is known to both players. On the 
th turn of the game (starting with 
) the following happens:
If is odd, Alice chooses a nonnegative real number 
such that
![\[x_1 + x_2 + \cdots + x_n \le \lambda n\]](http://latex.artofproblemsolving.com/a/1/9/a1980acef3a836807c505584b3159d385b662ba1.png)
If is even, Bazza chooses a nonnegative real number such that
![\[x_1^2 + x_2^2 + \cdots + x_n^2 \le n\]](http://latex.artofproblemsolving.com/8/1/e/81ef75d4416ca398f1f6972fae4f82cb3ea502a4.png)
If a player cannot choose a suitable , 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 for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
