InternGeometry：通过复杂度提升强化学习实现奥赛级几何大语言模型智能体

Computer Science > Artificial Intelligence arXiv:2512.10534 (cs) [Submitted on 11 Dec 2025 (v1), last revised 4 Sep 2026 (this version, v5)] Title:Achieving Olympiad-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning Authors:Haiteng Zhao, Junhao Shen, Yiming Zhang, Songyang Gao, Kuikun Liu, Tianyou Ma, Fan Zheng, Dahua Lin, Wenwei Zhang, Kai Chen View a PDF of the paper titled Achieving Olympiad-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning, by Haiteng Zhao and 9 other authors View PDF HTML (experimental) 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. Subjects: Artificial Intelligence (cs.AI) Cite as: arXiv:2512.10534 [cs.AI] (or arXiv:2512.10534v5 [cs.AI] for this version) https://doi.org/10.48550/arXiv.2512.10534 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Haiteng Zhao [view email] [v1] Thu, 11 Dec 2025 11:05:04 UTC (318 KB) [v2] Fri, 12 Dec 2025 13:43:03 UTC (655 KB) [v3] Thu, 5 Mar 2026 09:33:06 UTC (655 KB) [v4] Wed, 2 Sep 2026 14:03:03 UTC (655 KB) [v5] Fri, 4 Sep 2026 04:01:20 UTC (655 KB) Full-text links: Access Paper: View a PDF of the paper titled Achieving Olympiad-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning, by Haiteng Zhao and 9 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.AI new | recent | 2025-12 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)