Advanced version of Gemini with Deep Think officially achieves gold-medal standard at the International Mathematical Olympiad

The International Mathematical Olympiad (IMO) is the world’s most prestigious competition for young mathematicians, and has been held annually since 1959. Each country taking part is represented by six elite, pre-university mathematicians who compete to solve six exceptionally difficult problems in algebra, combinatorics, geometry, and number theory. Medals are awarded to the top half of contestants, with approximately 8% receiving a prestigious gold medal.

Recently, the IMO has also become an aspirational challenge for AI systems as a test of their advanced mathematical problem-solving and reasoning capabilities. Last year, Google DeepMind’s combined AlphaProof and AlphaGeometry 2 systems achieved the silver-medal standard, solving four out of the six problems and scoring 28 points. Making use of specialist formal languages, this breakthrough demonstrated that AI was beginning to approach elite human mathematical reasoning.

This year, we were amongst an inaugural cohort to have our model results officially graded and certified by IMO coordinators using the same criteria as for student solutions. Recognizing the significant accomplishments of this year’s student-participants, we’re now excited to share the news of Gemini’s breakthrough performance.

An advanced version of Gemini Deep Think solved five out of the six IMO problems perfectly, earning 35 total points, and achieving gold-medal level performance.

This achievement is a significant advance over last year’s breakthrough result. At IMO 2024, AlphaGeometry and AlphaProof required experts to first translate problems from natural language into domain-specific languages, such as Lean, and vice-versa for the proofs. It also took two to three days of computation. This year, our advanced Gemini model operated end-to-end in natural language, producing rigorous mathematical proofs directly from the official problem descriptions – all within the 4.5-hour competition time limit.

We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought.

To make the most of the reasoning capabilities of Deep Think, we additionally trained this version of Gemini on novel reinforcement learning techniques that can leverage more multi-step reasoning, problem-solving and theorem-proving data. We also provided Gemini with access to a curated corpus of high-quality solutions to mathematics problems, and added some general hints and tips on how to approach IMO problems to its instructions.

We will be making a version of this Deep Think model available to a set of trusted testers, including mathematicians, before rolling it out to Google AI Ultra subscribers.

Google DeepMind has ongoing collaborations with the mathematical community, but we are still only at the start of AI’s potential to contribute to mathematics. By teaching our systems to reason more flexibly and intuitively, we are getting closer to building AI that can solve more complex and advanced mathematics.

While our approach this year was based purely on natural language with Gemini, we also continue making progress on our formal systems, AlphaGeometry and AlphaProof. We believe agents that combine natural language fluency with rigorous reasoning - including verified reasoning in formal languages - will become invaluable tools for mathematicians, scientists, engineers, and researchers, helping us advance human knowledge on the path to AGI.