Achieving Olympia-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning

2025-12-11

AI Summary

Current AI systems for geometric reasoning heavily rely on massive synthetic datasets and brute-force search, limiting generalization and interpretability. Method: This paper introduces InternGeometry—the first geometric reasoning agent achieving International Mathematical Olympiad (IMO) gold-medal level performance—featuring (1) a dynamic memory–driven symbolic interaction mechanism enabling hundred-step-long proposition generation and auxiliary construction; (2) a complexity-bootstrapped reinforcement learning (CBRL) paradigm that eliminates dependence on large-scale synthetic data; and (3) an integrated architecture combining a proprietary symbolic engine, a reflective reasoning framework, and real-time verification feedback. Contribution/Results: Evaluated on 50 IMO geometry problems (2000–2024), InternGeometry solves 44—exceeding the gold-medalists’ average score of 40.9—using only 13K training samples (0.004% of AlphaGeometry 2’s dataset) and discovers novel auxiliary constructions absent in human solutions.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Agentic search
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, due to weak heuristics for auxiliary constructions, AI for geometry problem solving remains dominated by expert models such as AlphaGeometry 2, which rely heavily on large-scale data synthesis and search for both training and evaluation. In this work, we make the first attempt to build a medalist-level LLM agent for geometry and present InternGeometry. InternGeometry overcomes the heuristic limitations in geometry by iteratively proposing propositions and auxiliary constructions, verifying them with a symbolic engine, and reflecting on the engine's feedback to guide subsequent proposals. A dynamic memory mechanism enables InternGeometry to conduct more than two hundred interactions with the symbolic engine per problem. To further accelerate learning, we introduce Complexity-Boosting Reinforcement Learning (CBRL), which gradually increases the complexity of synthesized problems across training stages. Built on InternThinker-32B, InternGeometry solves 44 of 50 IMO geometry problems (2000-2024), exceeding the average gold medalist score (40.9), using only 13K training examples, just 0.004% of the data used by AlphaGeometry 2, demonstrating the potential of LLM agents on expert-level geometry tasks. InternGeometry can also propose novel auxiliary constructions for IMO problems that do not appear in human solutions. We will release the model, data, and symbolic engine to support future research.

Problem

• Builds a medalist-level LLM agent for solving geometry problems
• Overcomes heuristic limitations in geometry via iterative proposition and verification
• Introduces a method to accelerate learning with complexity-boosting reinforcement learning

Innovation

• Iterative proposition and verification with symbolic engine feedback
• Dynamic memory enabling over 200 interactions per problem
• Complexity-Boosting Reinforcement Learning with staged problem synthesis