AlphaProof and AlphaGeometry 2 reach IMO silver-medal standard

Background

The International Mathematical Olympiad (IMO) is a competition in which students solve mathematical problems and present rigorous written arguments. Its problems are designed to require proof and insight rather than routine calculation. On 25 July 2024, Google DeepMind announced results from two of its systems, AlphaProof and AlphaGeometry 2, on problems from the 2024 IMO.^[1]

The announcement described an evaluation in which the systems attempted IMO problems, with the problems translated into formal language by human experts.^[1] That detail matters because a formal language is a notation in which each step of a proof can be written precisely and checked mechanically. The translation step therefore formed part of the pipeline: the human experts converted the original problem statements into a form the systems could work with.^[1]

What happened

Google DeepMind said the systems solved four of the six problems.^[1] AlphaProof solved two algebra problems and one number-theory problem, while AlphaGeometry 2 solved the geometry problem.^[1] The two combinatorics problems remained unsolved.^[1] Across the six problems, the solutions scored 28 of 42 points.^[1] The gold threshold was 29 points, so the result fell just below gold and matched silver-medal level.^[1] Google DeepMind said this reached the same level as a silver medallist for the first time.^[1]

Why the result was notable

The IMO is used as a public test of mathematical problem-solving because its problems require constructing arguments, not merely producing numerical answers. A system that can solve such problems must produce a chain of reasoning that can be checked. In this evaluation, the use of formal language gave the systems a precise target for proof construction, and the human translation of the problems supplied the starting point.^[1]

The score also made the boundary of the result clear. Four problems were solved, including algebra, number theory and geometry.^[1] Two combinatorics problems were not solved.^[1] The total of 28 points placed the run below the 29-point gold threshold but at the level Google DeepMind described as silver-medal standard.^[1] Because the problems were translated by human experts, the reported result concerned problem-solving after that translation, not the automatic translation of the original statements.^[1]

What the announcement reported

The announcement gave a concrete comparison with human competition scoring: 28 of 42 points, just short of the gold threshold, and a silver-medal level as stated by Google DeepMind.^[1] It also separated the systems by problem type: AlphaProof handled the two algebra problems and the number-theory problem, while AlphaGeometry 2 handled the geometry problem.^[1] The two combinatorics problems stayed outside the solved set.^[1] Those details define the scope of the reported result and the specific gap that remained in this evaluation.