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IMO2025Q5

In this video I solve Question 5 of the 2025 International Mathematical Olympiad in real time. To get the most out of the video, you might like to try the question for yourself before watching. The question asks the following. Alice and Bazza play the following game. They are given a real number λ and then take turns choosing non-negative real numbers x_1, x_2, x_3, ... and so on, with Alice starting. Each time Alice chooses a number x_n, she must do so in such a way that x_1 + ... + x_n is at most λn, and each time Bazza chooses a number n, he must do so in such a way that x_1^2 + ... + x_n^2 is at most n. If either player cannot choose such an x_n, then the other player wins, and if the game goes on for ever, then it is a draw. For which values of λ does Alice have a winning strategy and for which values of λ does Bazza have a winning strategy? I had a particular interest in this question because one of the two setters, Leonardo Franchi, is due to become a PhD student of mine next October. His co-setter was Massimiliano Foschi. Congratulations to the two of them on coming up with a very nice problem and on having it selected for the competition.