Published on July 25, 2024 3:58 PM GMT
Google DeepMind reports on a system for solving mathematical problems that allegedly is able to give complete solutions to four of the six problems on the 2024 IMO, putting it near the top of the silver-medal category.
Well, actually, two systems for solving mathematical problems: AlphaProof, which is more general-purpose, and AlphaGeometry, which is specifically for geometry problems. (This is AlphaGeometry 2; they reported earlier this year on a previous version of AlphaGeometry.)
AlphaProof works in the "obvious" way: an LLM generates formalized versions of the things to be proved, and generates candidate next steps which are checked using a formal proof-checking system, in this case Lean. One not-so-obvious thing, though: "The training loop was also applied during the contest, reinforcing proofs of self-generated variations of the contest problems until a full solution could be found."
AlphaGeometry is similar but uses something specialized for (I think) Euclidean planar geometry problems in place of Lean. The previous version of AlphaGeometry allegedly already performed at gold-medal IMO standard; they don't say anything about whether that version was already able to solve the 2024 IMO problem that was solved using AlphaGeometry 2.
AlphaProof was able to solve questions 1, 2, and 6 on this year's IMO (two algebra, one number theory). It produces Lean-formalized proofs. AlphaGeometry 2 was able to solve question 4 (plane geometry). It produces proofs in its own notation.
The solutions found by the Alpha... systems are at https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/imo-2024-solutions/index.html. (There are links in the top-of-page navbar to solutions to the individual problems.)
(If you're curious about the IMO questions or want to try them yourself before looking at the machine-generated proofs, you can find them -- and those for previous years -- at https://www.imo-official.org/problems.aspx.)
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