IMO 2025 Problem 5: the inekoalaty game
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Fix λ>0. On odd turn n Alice chooses xₙ≥0 with Σᵢ₌₁ⁿxᵢ≤λn; on even turn n Bazza chooses xₙ≥0 with Σᵢ₌₁ⁿxᵢ²≤n. All moves are visible. A player unable to move loses; infinite play has no winner. Determine the λ values for which each player has a winning strategy.
Why it matters and what counts as progress
Why it matters: It asks how alternating linear and quadratic resource constraints determine winning strategies and the transition between them.
What counts as progress: Winning strategies for the complete parameter ranges, with a separate analysis of any boundary value and the possibility of infinite play.
How the problem developed
- 16 Jul 2025 · Contest problem [1]
IMO jury The problem was set on this day of the official IMO 2025 contest.
Google DeepMind and Gemini Deep Think DeepMind reported an officially graded five-of-six result (35/42); its published solution collection includes a proof for this problem.
AI contributions 2
Aristotle formalizes five IMO 2025 solutions
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
Gemini Deep Think solves five IMO 2025 problems
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
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