用强化学习让大模型自主解决国际奥数几何题，仅用少量数据超越人类金牌水平。
Achieving Olympia-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning
- 通过动态记忆和符号引擎反馈，迭代生成几何辅助线与命题
- 在50道奥数几何题中解出44道，超过金牌平均分40.9分
- 仅用13,000个训练样本，数据量仅为AlphaGeometry 2的0.004%
大型语言模型（LLM）代理在数学问题求解方面表现出色，甚至能在形式化证明系统辅助下解决国际数学奥林匹克（IMO）级别的题目。然而，由于对辅助构造缺乏有效启发式方法，几何问题求解仍由专家模型如AlphaGeometry 2主导，其依赖大规模数据合成与搜索进行训练和评估。本文首次尝试构建达到奖牌水平的几何专用LLM代理，提出InternGeometry。该模型通过迭代提出命题与辅助构造，利用符号引擎验证并基于反馈反思，指导后续生成。动态记忆机制使其每道题可与符号引擎交互超200次。为进一步加速学习，引入复杂度提升强化学习（CBRL），逐步增加合成题目的复杂性。基于InternThinker-32B，InternGeometry成功解决50道IMO几何题中的44道（2000–2024年），得分超过平均金牌线40.9，仅使用13,000个训练样本，不足AlphaGeometry 2所用数据的0.004%，展现出大模型代理在专家级几何任务上的潜力。此外，该模型还能为未见于人类解法的题目提出新颖辅助构造。
原文摘要 · Abstract (English)
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.
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