Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2

TL;DR

AlphaGeometry2 (AG2) is an advanced AI system that surpasses gold medalists in Olympiad geometry problem-solving by extending language capabilities, improving search architecture, and achieving high coverage and success rates on IMO problems.

Contribution

The paper introduces AG2, a significantly improved geometry-solving system with extended problem coverage, enhanced search architecture, and higher success rates, surpassing previous versions and human gold medalists.

Findings

Coverage rate increased from 66% to 88% on IMO problems.

Overall solving rate improved from 54% to 84%.

AG2 achieved silver-medal level at IMO 2024.

Abstract

We present AlphaGeometry2 (AG2), a significantly improved version of AlphaGeometry introduced in (Trinh et al., 2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with support for non-constructive problems, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AG2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that enables effective communication between search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AG to 84% on all geometry problems over the last 25 years, compared to 54% previously. AG2 was also part of the system that achieved the silver-medal standard at IMO 2024 https://deepmind.google/blog/ai-solves-imo-problems-at-silver-medal-level/. Finally, we report progress towards using AG2 as a part of a fully automated system that reliably solves geometry problems from natural language input. Code: https://github.com/google-deepmind/alphageometry2.