Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a highly efficient method for geometry theorem proving that runs entirely on CPUs without relying on neural network-based inference. Our initial study shows that a simple random strategy for adding auxiliary points can achieve silver-medal level human performance on IMO. Building on this, we propose HAGeo, a Heuristic-based method for adding Auxiliary constructions in Geometric deduction that solves 28 of 30 problems on the IMO-30 benchmark, achieving gold-medal level performance and surpassing AlphaGeometry, a competitive neural network-based approach, by a notable margin. To evaluate our method and existing approaches more comprehensively, we further construct HAGeo-409, a benchmark consisting of 409 geometry problems with human-assessed difficulty levels. Compared with the widely used IMO-30, our benchmark poses greater challenges and provides a more precise evaluation, setting a higher bar for geometry theorem proving.

Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated theorem discovery and proving, especially in geometry for its combination of numerical and spatial elements. We introduce TongGeometry, a Euclidean geometry system supporting tree-search-based guided problem proposing and solving. The efficient geometry system establishes the most extensive repository of geometry theorems to date: within the same computational budget as the existing state-of-the-art, TongGeometry discovers 6.7 billion geometry theorems requiring auxiliary constructions, including 4.1 billion exhibiting geometric symmetry. Among them, 10 theorems were proposed to regional mathematical olympiads with 3 of TongGeometry's proposals selected in real competitions, earning spots in a national team qualifying exam or a top civil olympiad in China and the US. Guided by fine-tuned large language models, TongGeometry solved all International Mathematical Olympiad geometry in IMO-AG-30, outperforming gold medalists for the first time. It also surpasses the existing state-of-the-art across a broader spectrum of olympiad-level problems. The full capabilities of the system can be utilized on a consumer-grade machine, making the model more accessible and fostering widespread democratization of its use. By analogy, unlike existing systems that merely solve problems like students, TongGeometry acts like a geometry coach, discovering, presenting, and proving theorems.

DeepMind's AlphaProof AI can tackle a range of mathematical problems An AI from Google DeepMind has achieved a silver medal score at this year's International Mathematical Olympiad (IMO), the first time any AI has made it to the podium. The IMO is considered the world's most prestigious competition for young mathematicians. Correctly answering its test questions requires mathematical ability that AI systems typically lack. In January, Google DeepMind demonstrated AlphaGeometry, an AI system that could answer some IMO geometry questions as well as humans. However, this was not from a live competition, and it couldn't answer questions from other mathematical disciplines, such as number theory, algebra and combinatorics, which is necessary to win an IMO medal.

Several years before ChatGPT began jibber-jabbering away, Google developed a very different kind of artificial intelligence program called AlphaGo that learned to play the board game Go with superhuman skill through tireless practice. Researchers at the company have now published research that combines the abilities of a large language model (the AI behind today's chatbots) with those of AlphaZero, a successor to AlphaGo also capable of playing chess, to solve very tricky mathematical proofs. Their new Frankensteinian creation, dubbed AlphaProof, has demonstrated its prowess by tackling several problems from the 2024 International Math Olympiad (IMO), a prestigious competition for high school students. AlphaProof uses the Gemini large language model to convert naturally phrased math questions into a programming language called Lean. This provides the training fodder for a second algorithm to learn, through trial and error, how to find proofs that can be confirmed as correct.

Proving geometric theorems constitutes a hallmark of visual reasoning combining both intuitive and logical skills. Therefore, automated theorem proving of Olympiad-level geometry problems is considered a notable milestone in human-level automated reasoning. The introduction of AlphaGeometry, a neuro-symbolic model trained with 100 million synthetic samples, marked a major breakthrough. It solved 25 of 30 International Mathematical Olympiad (IMO) problems whereas the reported baseline based on Wu's method solved only ten. In this note, we revisit the IMO-AG-30 Challenge introduced with AlphaGeometry, and find that Wu's method is surprisingly strong. Wu's method alone can solve 15 problems, and some of them are not solved by any of the other methods. This leads to two key findings: (i) Combining Wu's method with the classic synthetic methods of deductive databases and angle, ratio, and distance chasing solves 21 out of 30 methods by just using a CPU-only laptop with a time limit of 5 minutes per problem. Essentially, this classic method solves just 4 problems less than AlphaGeometry and establishes the first fully symbolic baseline strong enough to rival the performance of an IMO silver medalist. (ii) Wu's method even solves 2 of the 5 problems that AlphaGeometry failed to solve. Thus, by combining AlphaGeometry with Wu's method we set a new state-of-the-art for automated theorem proving on IMO-AG-30, solving 27 out of 30 problems, the first AI method which outperforms an IMO gold medalist.

An AI from Google DeepMind can solve some International Mathematical Olympiad (IMO) questions on geometry almost as well as the best human contestants. How does ChatGPT work and do AI-powered chatbots "think" like us? "The results of AlphaGeometry are stunning and breathtaking," says Gregor Dolinar, the IMO president. "It seems that AI will win the IMO gold medal much sooner than was thought even a few months ago." The IMO, aimed at secondary school students, is one of the most difficult maths competitions in the world. Answering questions correctly requires mathematical creativity that AI systems have long struggled with.

Solving mathematics problems requires logical reasoning, something that most current AI models aren't great at. This demand for reasoning is why mathematics serves as an important benchmark to gauge progress in AI intelligence, says Wang. DeepMind's program, named AlphaGeometry, combines a language model with a type of AI called a symbolic engine, which uses symbols and logical rules to make deductions. Language models excel at recognizing patterns and predicting subsequent steps in a process. However, their reasoning lacks the rigor required for mathematical problem-solving. The symbolic engine, on the other hand, is based purely on formal logic and strict rules, which allows it to guide the language model toward rational decisions.