AlphaGeometry solves olympiad geometry problems near gold-medal level

Google DeepMind published AlphaGeometry, a system that solved 25 of 30 benchmark geometry problems drawn from International Mathematical Olympiad-level difficulty, within the time limits the competition allows. The average human gold medallist across those 30 problems solved 25.9. The best prior automated system had managed only 10.

AlphaGeometry combined two components: a neural language model that proposes which auxiliary construction — an added point, line or circle — might help solve a diagram, and a symbolic deduction engine that applies formal logical rules to derive conclusions and check whether a proof is valid. DeepMind described the language model as supplying “fast, intuitive” suggestions that the symbolic engine then verified through slower, rule-based reasoning — a division of labour explicitly modelled on the distinction between intuition and deliberate reasoning in human problem-solving.

A notable feature of the approach was how it was trained: rather than relying on a limited supply of real olympiad problems and human-written solutions, DeepMind generated around 100 million synthetic geometry theorems and proofs and trained the system on those, avoiding the data bottleneck that constrains most specialised reasoning systems. DeepMind open-sourced both the code and the trained model.

Geometry problems are one of the more tractable Olympiad categories for this kind of hybrid search-and-verify approach, since geometric statements can be formally encoded and mechanically checked in a way informal algebra or combinatorics problems often cannot. The result was nonetheless read as evidence that neuro-symbolic methods — pairing a learned model with a formal verifier — could match elite human performance on a well-defined mathematical task requiring genuine construction rather than lookup. DeepMind returned to the approach eighteen months later, pairing a successor system, AlphaGeometry 2, with the reinforcement-learning system AlphaProof to reach silver-medal standard across the full range of 2024 IMO problems.