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文章背景与核心概要

本文介绍了 InternGeometry，这是首个专为几何学设计的、达到奥林匹克金牌水平的大语言模型（LLM）智能体。长期以来，几何问题求解因需要复杂的辅助线构建和启发式搜索，一直由 AlphaGeometry 2 等依赖大规模数据合成的专用专家模型主导。而 InternGeometry 基于 InternThinker-32B 基础模型构建，通过迭代式地提出命题与辅助线、利用符号引擎进行验证，并根据引擎反馈进行自我反思与调整，成功突破了传统几何推理的启发式限制。

该研究的核心创新在于引入了复杂度提升强化学习（Complexity-Boosting Reinforcement Learning, CBRL）范式，在训练的不同阶段逐步提高合成问题的复杂度，从而加速模型对复杂几何任务的掌握。凭借仅 13K 的训练样本（仅为 AlphaGeometry 2 数据量的 0.004%），InternGeometry 成功解决了 2000 年至 2024 年间 50 道 IMO 几何题中的 44 道，超越了人类金牌得主 40.9 的平均得分。这一成果充分展现了大语言模型智能体在专家级几何任务中的巨大潜力，甚至能够为 IMO 问题生成人类解法中未曾出现的全新辅助线。

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

arXiv: 2512.10534 [cs.AI]
Authors: Haiteng Zhao, Junhao Shen, Yiming Zhang, Songyang Gao, Kuikun Liu, Tianyou Ma, Fan Zheng, Dahua Lin, Wenwei Zhang, Kai Chen
Submitted: December 11, 2025; Last revised: September 4, 2026 (v5)

Executive Summary

大语言模型（LLM）智能体已经展现出强大的数学推理能力，有时甚至能在形式化证明系统的辅助下解决国际数学奥林匹克（IMO）级别的问题。然而，传统的几何问题求解一直由 AlphaGeometry 2 等专用专家模型主导，这些模型严重依赖海量的数据合成和广泛的搜索策略。

本文介绍了 InternGeometry，这是首个专为几何学设计的金牌级 LLM 智能体。InternGeometry 基于 InternThinker-32B 基础模型构建，通过提出命题和辅助线、通过符号引擎进行验证、并根据反馈精炼其解题方法的迭代过程，克服了传统的启发式限制。

核心亮点：

  • 卓越性能： 解决了 2000 年至 2024 年间 50 道 IMO 几何题中的 44 道，超过了人类金牌得主 40.9 的平均得分。
  • 数据高效： 仅使用 13K 训练样本便达到了这一性能——仅占 AlphaGeometry 2 使用的数据量的 0.004%。
  • 新颖的辅助线构建： 能够为 IMO 问题生成人类解法中未曾出现过的全新辅助线。
  • 复杂度提升强化学习（CBRL）： 一种全新的学习范式，在训练阶段逐步提升合成问题的复杂度，以加速模型精通几何任务。

Large language model (LLM) agents have demonstrated powerful mathematical reasoning capabilities, sometimes tackling International Mathematical Olympiad (IMO) level problems with formal proof systems. However, geometry problem-solving has traditionally remained dominated by specialized expert models like AlphaGeometry 2, which rely heavily on massive data synthesis and extensive search strategies.

This paper introduces InternGeometry, the first medalist-level LLM agent designed specifically for geometry. Built upon the InternThinker-32B foundation, InternGeometry overcomes traditional heuristic limitations through an iterative process of proposing propositions and auxiliary constructions, verifying them via a symbolic engine, and refining its approach based on feedback.

Key Highlights:

  • Exceptional Performance: Solves 44 out of 50 IMO geometry problems from 2000 to 2024, surpassing the average human gold medalist score of 40.9.
  • Data Efficiency: Achieves this performance using only 13K training examples—just 0.004% of the data volume utilized by AlphaGeometry 2.
  • Novel Auxiliary Constructions: Capable of generating entirely original auxiliary constructions for IMO problems that do not appear in human solution sets.
  • Complexity-Boosting Reinforcement Learning (CBRL): A novel learning paradigm that incrementally scales up the complexity of synthesized problems across training stages to accelerate mastery.

Abstract

大语言模型（LLM）智能体展现出了强大的数学问题求解能力，甚至可以在形式化证明系统的辅助下解决国际数学奥林匹克（IMO）级别的问题。然而，由于缺乏针对辅助线构建的强有力启发式方法，面向几何问题求解的 AI 长期以来一直被 AlphaGeometry 2 等专家模型所主导，这些模型在训练和评估阶段都严重依赖大规模的数据合成与搜索。

在这项工作中，我们进行了首次尝试，旨在构建一个几何领域的金牌级 LLM 智能体，并提出了 InternGeometry。InternGeometry 通过迭代式地提出命题与辅助线、利用符号引擎进行验证，并反思引擎的反馈来指导后续的假设生成，从而克服了几何学中的启发式限制。动态内存机制使 InternGeometry 能够针对单个问题与符号引擎进行两百多次交互。

为了进一步加速学习，我们引入了复杂度提升强化学习（Complexity-Boosting Reinforcement Learning, CBRL），该方法在训练的不同阶段逐步提高合成问题的复杂度。基于 InternThinker-32B 构建的 InternGeometry 解决了 2000 至 2024 年间 50 道 IMO 几何题中的 44 道，超过了金牌得主的平均得分（40.9 分）；其训练过程仅使用了 13K 示例，仅为 AlphaGeometry 2 使用的数据量的 0.004%，充分展示了 LLM 智能体在专家级几何任务中的潜力。此外，InternGeometry 还能为 IMO 问题生成人类解法中从未出现过的全新辅助线。

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.

References & Additional Resources