IMO 2024 Problem 1: divisible floor sums
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Determine all real α for which Σₖ₌₁ⁿ⌊kα⌋ is divisible by n for every positive integer n. Here ⌊x⌋ is the greatest integer at most x.
Why it matters and what counts as progress
Why it matters: The problem turns an infinite family of floor-sum divisibility conditions into a complete classification of a real parameter.
What counts as progress: A proof classifying every admissible real α, including sufficiency of the even integers and exclusion of all other reals; a faithful formalization is a further advance.
How the problem developed
- 16 Jul 2024 · Contest problem [1]
IMO jury The problem appeared in the official IMO 2024 contest on this contest day.
Google DeepMind AlphaProof and AlphaGeometry solved four of six IMO 2024 problems; DeepMind published a proof page for this problem. Its problem page documents which formalization and proof system were used.
AI contribution
Olympiad-level formal mathematical reasoning with reinforcement learning
DeepMind reports Lean proofs for IMO 2024 problems 1, 2, and 6, and publishes the specific proof artifacts.
Sources
Website timeline
Publication dates and editorial changes.