Gold-Medal-Level Olympiad Geometry Solving with Efficient Heuristic Auxiliary Constructions

2025-11-26

AI Summary

Addressing the long-standing challenge of automated theorem proving for International Mathematical Olympiad (IMO)-level Euclidean geometry problems, this paper introduces HAGeo—a lightweight, purely CPU-based, neural-network-free symbolic reasoning framework. Its core innovation is heuristic-assisted geometric construction, integrating rule-driven search, stochastic auxiliary point generation, and a formally verifiable geometric reasoning engine to produce end-to-end interpretable proofs. Evaluated on the IMO-30 benchmark, HAGeo solves 28 out of 30 problems—achieving gold-medal performance and substantially outperforming neural approaches such as AlphaGeometry. To advance standardized evaluation, we further introduce HAGeo-409, a rigorously curated benchmark comprising 409 original, high-difficulty geometry problems. This work establishes, for the first time, that compact, interpretable symbolic methods can match or exceed state-of-the-art neural systems in elite-level geometric reasoning tasks.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Heuristic Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

Problem

• Automated theorem proving for IMO-level Euclidean geometry
• Efficient heuristic method for adding auxiliary constructions
• New benchmark for comprehensive geometry problem evaluation

Innovation

• Heuristic-based auxiliary construction method
• CPU-only geometry theorem proving
• New benchmark HAGeo-409 for evaluation