# Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2

arXiv: https://arxiv.org/abs/2502.03544 · HTML: https://mlsanity.com/paper/2502.03544/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 42 references that are in the map.
- First posted on arXiv 5 Feb 2025.

## Builds on (most cited first, 22 of 22)

- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Large Language Models Cannot Self-Correct Reasoning Yet](https://mlsanity.com/paper/2310.01798/) (2023)
- [Teaching Large Language Models to Self-Debug](https://mlsanity.com/paper/2304.05128/) (2023)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Visual Sketchpad: Sketching as a Visual Chain of Thought for Multimodal Language Models](https://mlsanity.com/paper/2406.09403/) (2024)
- [Autoformalization with Large Language Models](https://mlsanity.com/paper/2205.12615/) (2022)
- [On the Self-Verification Limitations of Large Language Models on Reasoning and Planning Tasks](https://mlsanity.com/paper/2402.08115/) (2024)
- [Tokenization counts: the impact of tokenization on arithmetic in frontier LLMs](https://mlsanity.com/paper/2402.14903/) (2024)
- [Multilingual Mathematical Autoformalization](https://mlsanity.com/paper/2311.03755/) (2023)
- [Reliable Evaluation and Benchmarks for Statement Autoformalization](https://mlsanity.com/paper/2406.07222/) (2024)
- [Autoformalizing Euclidean Geometry](https://mlsanity.com/paper/2405.17216/) (2024)
- [Proving Olympiad Algebraic Inequalities without Human Demonstrations](https://mlsanity.com/paper/2406.14219/) (2024)
- [Proposing and solving olympiad geometry with guided tree search](https://mlsanity.com/paper/2412.10673/) (2024)
- [Beyond Lines and Circles: Unveiling the Geometric Reasoning Gap in Large Language Models](https://mlsanity.com/paper/2402.03877/) (2024)
- [Wu's Method can Boost Symbolic AI to Rival Silver Medalists and AlphaGeometry to Outperform Gold Medalists at IMO Geometry](https://mlsanity.com/paper/2404.06405/) (2024)
- [T-FREE: Subword Tokenizer-Free Generative LLMs via Sparse Representations for Memory-Efficient Embeddings](https://mlsanity.com/paper/2406.19223/) (2024)
- [Automatically Building Diagrams for Olympiad Geometry Problems](https://mlsanity.com/paper/2012.02590/) (2020)
- [Do Large Language Models Truly Understand Geometric Structures?](https://mlsanity.com/paper/2501.13773/) (2025)

## Built on by (most cited first, 30 of 97)

- [From System 1 to System 2: A Survey of Reasoning Large Language Models](https://mlsanity.com/paper/2502.17419/) (2025)
- [Goedel-Prover-V2: Scaling Formal Theorem Proving with Scaffolded Data Synthesis and Self-Correction](https://mlsanity.com/paper/2508.03613/) (2025)
- [Seed-Prover: Deep and Broad Reasoning for Automated Theorem Proving](https://mlsanity.com/paper/2507.23726/) (2025)
- [DeepSeekMath-V2: Towards Self-Verifiable Mathematical Reasoning](https://mlsanity.com/paper/2511.22570/) (2025)
- [Winning Gold at IMO 2025 with a Model-Agnostic Verification-and-Refinement Pipeline](https://mlsanity.com/paper/2507.15855/) (2025)
- [AutoHarness: improving LLM agents by automatically synthesizing a code harness](https://mlsanity.com/paper/2603.03329/) (2026)
- [Towards Robust Mathematical Reasoning](https://mlsanity.com/paper/2511.01846/) (2025)
- [Solver-Informed RL: Grounding Large Language Models for Authentic Optimization Modeling](https://mlsanity.com/paper/2505.11792/) (2025)
- [Feedback Descent: Open-Ended Text Optimization via Pairwise Comparison](https://mlsanity.com/paper/2511.07919/) (2025)
- [ReflAct: World-Grounded Decision Making in LLM Agents via Goal-State Reflection](https://mlsanity.com/paper/2505.15182/) (2025)
- [Ax-Prover: A Deep Reasoning Agentic Framework for Theorem Proving in Mathematics and Quantum Physics](https://mlsanity.com/paper/2510.12787/) (2025)
- [ICL-Router: In-Context Learned Model Representations for LLM Routing](https://mlsanity.com/paper/2510.09719/) (2025)
- [Cognitive Foundations for Reasoning and Their Manifestation in LLMs](https://mlsanity.com/paper/2511.16660/) (2025)
- [Critique of World Model](https://mlsanity.com/paper/2507.05169/) (2025)
- [Mathematical Proof as a Litmus Test: Revealing Failure Modes of Advanced Large Reasoning Models](https://mlsanity.com/paper/2506.17114/) (2025)
- [Enhancing the Geometric Problem-Solving Ability of Multimodal LLMs via Symbolic-Neural Integration](https://mlsanity.com/paper/2504.12773/) (2025)
- [From Problem-Solving to Teaching Problem-Solving: Aligning LLMs with Pedagogy using Reinforcement Learning](https://mlsanity.com/paper/2505.15607/) (2025)
- [TrustGeoGen: Formal-Verified Data Engine for Trustworthy Multi-modal Geometric Problem Solving](https://mlsanity.com/paper/2504.15780/) (2025)
- [Bottom-up Domain-specific Superintelligence: A Reliable Knowledge Graph is What We Need](https://mlsanity.com/paper/2507.13966/) (2025)
- [The Art of Scaling Test-Time Compute for Large Language Models](https://mlsanity.com/paper/2512.02008/) (2025)
- [Pi-GPS: Enhancing Geometry Problem Solving by Unleashing the Power of Diagrammatic Information](https://mlsanity.com/paper/2503.05543/) (2025)
- [Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics](https://mlsanity.com/paper/2605.13171/) (2026)
- [GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization](https://mlsanity.com/paper/2506.07160/) (2025)
- [Autonomous Research Agents: A Survey of AI Scientists and the Verification Gap](https://mlsanity.com/paper/2608.05179/) (2026)
- [Towards Geometry Problem Solving in the Large Model Era: A Survey](https://mlsanity.com/paper/2506.02690/) (2025)
- [PuzzleWorld: A Benchmark for Multimodal, Open-Ended Reasoning in Puzzlehunts](https://mlsanity.com/paper/2506.06211/) (2025)
- [Riemann-Bench: A Benchmark for Moonshot Mathematics](https://mlsanity.com/paper/2604.06802/) (2026)
- [Reliable Proof Generation with LLMs via Analogical Retrieval and Symbolic Verification: A Case Study in Euclidean Geometry](https://mlsanity.com/paper/2505.14479/) (2025)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [Semantic Voting: A Self-Evaluation-Free Approach for Efficient LLM Self-Improvement on Unverifiable Open-ended Tasks](https://mlsanity.com/paper/2509.23067/) (2025)

## Abstract (arXiv)

We present AlphaGeometry2 (AG2), a significantly improved version of AlphaGeometry introduced in (Trinh et al., 2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with support for non-constructive problems, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AG2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that enables effective communication between search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AG to 84% on all geometry problems over the last 25 years, compared to 54% previously. AG2 was also part of the system that achieved the silver-medal standard at IMO 2024 https://deepmind.google/blog/ai-solves-imo-problems-at-silver-medal-level/. Finally, we report progress towards using AG2 as a part of a fully automated system that reliably solves geometry problems from natural language input. Code: https://github.com/google-deepmind/alphageometry2.