Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2

2025-02-05

AI Summary

AlphaGeometry exhibits limited performance on International Mathematical Olympiad (IMO) geometry problems, constrained by insufficient formal language coverage and inability to handle complex geometric transformations (e.g., rigid motions) and linear constraints (e.g., angle/distance/proportion equations). Method: We propose AlphaGeometry2, featuring: (1) an extended geometric symbolic language supporting rigid-body transformations and linear equation modeling over angles, ratios, and distances; (2) a novel multi-search-tree architecture with collaborative knowledge sharing to enhance reasoning efficiency; and (3) integration of the Gemini large language model to strengthen natural language understanding and synthetic data generation. Results: Evaluated on all IMO geometry problems from 2000–2024, AlphaGeometry2 achieves 88% problem coverage (up from 66%) and 84% overall solving rate (up from 54%), attaining IMO 2024 silver-medal performance—marking the first end-to-end fully automated geometric theorem prover to demonstrate practical utility at real-world competition scale.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We present AlphaGeometry2, 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 harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, 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 AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for $ extit{all}$ geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.

Problem

• AlphaGeometry2 surpasses gold medalists in Olympiad geometry.
• Enhances solving complex geometry with object movements and equations.
• Boosts solving rate to 84% for geometry problems.

Innovation

• Enhanced AlphaGeometry language
• Gemini architecture for modeling
• Knowledge-sharing mechanism integration