International Mathematical Olympiad (IMO) is a global recognized competition that challenges high school students with complex mathematical problems. In its four categories, geometry is most consistent in structure, making it more accessible and well suited to fundamental logic research. Automatic Geometry Problem-Samadhan has traditionally followed two primary approaches: algebraic methods, such as Wu’s method, field method, and grobner base, and synthetic technology, including cut databases and full angle method. The latter aligns more closely with human logic and is particularly valuable for comprehensive research applications.
Previous research introduced alphageometry (AG1) to a neuro-sembolic system designed to solve the imo geometry problems by integrating a language model with a symbolic argument engine. From 2000 to 2024, AG1 achieved 54% success rate on issues, marking a significant step in automatic problems. However, its performance was interrupted by its domain-specific language boundaries, its symbolic engine efficiency, and the ability of its early language model. These obstacles stopped the AG1 from crossing their current accuracy despite their promising approach.
Alphageometry2 (AG2) is a major advancement on its predecessor, which exceeds the problem-composition capabilities of an average IMO gold medalist. Researchers at Google Deepmind, The University of Cambridge, Georgia Tech and Brown University expanded their domain language to handle complex geometric concepts, improving coverage of IMO problems from 66% to 88%. The AG2 integrates a Gemini-based language model, a more efficient symbolic engine and a novel search algorithm with knowledge sharing. These enhancers increase the rate of solution to 84% on IMO geometry problems from 2000-2024. Additionally, the AG2 leads to a completely automated system that explains problems from the natural language.
The AG2 expands the AG1 domain language by introducing additional Bills to address boundaries in linear equations, movement and general geometric problems. This increases coverage from 66% to 88% of IMO geometry problems (2000–2024). The AG2 supports new problems types, such as locos problems, and the diagram improves the formalities by allowing the points to define the points. The automatic formalities aided by the Foundation model translates the natural language problems into AG Syntax. Diagram generations employ two-step adaptation method for non-construction problems. AG2 also strengthens its symbolic engine, DDAR, rapidly and more efficient to stop cuts, enhances evidence search abilities.
Alphageometry2 receives a high plow rate on IMO geometry problems from 2000-2024, the IMO -g-50 solves 42 out of 50 in benchmarks, crossing an average gold medalist. It also solves all 30 most difficult formal IMO shortlist problems. Performance improves rapidly by solving 27 problems after 250 training stages. Ablation studies reveal optimal estimates settings. Some issues in DDAR remain unresolved due to unexpected conditions or lack of advanced geometry techniques. Experts find its solutions highly creative. Despite the boundaries, alphageometry2 outperforms AG1 and other systems, performing state-of-the-art capabilities in automatic problems.
Finally, alphageometry2 improves its predecessor by incorporating a more advanced language model, a enhanced symbolic engine and a novel proof search algorithm. It receives the rate of 84% solution on 2000–2024 imo geometry problems, which crosses the previous 54%. Studies show that language models can generate complete evidence without external devices, and various training approaches receive complementary skills. Challenges remain, including boundaries in handling inequalities and convertible points. Future work will focus on subprolem decomposition, reinforcement learning and refining auto-formalization for more reliable solutions in future work. The purpose of continuous reforms is to create a completely automated system to solve geometry problems efficiently.
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Sana Hasan, a counseling intern and double degree student at Marktekpost in IIT Madras, is emotional about implementing technology and AI to resolve real -world challenges. With a keen interest in solving practical problems, he brings a new approach to the intersection of AI and real -life solutions.
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