Simon Willison’s Weblog

OpenAI’s gold medal performance on the International Math Olympiad. This feels notable to me. OpenAI research scientist Alexander Wei:

I’m excited to share that our latest @OpenAI experimental reasoning LLM has achieved a longstanding grand challenge in AI: gold medal-level performance on the world’s most prestigious math competition—the International Math Olympiad (IMO).

We evaluated our models on the 2025 IMO problems under the same rules as human contestants: two 4.5 hour exam sessions, no tools or internet, reading the official problem statements, and writing natural language proofs. [...]

Besides the result itself, I am excited about our approach: We reach this capability level not via narrow, task-specific methodology, but by breaking new ground in general-purpose reinforcement learning and test-time compute scaling.

In our evaluation, the model solved 5 of the 6 problems on the 2025 IMO. For each problem, three former IMO medalists independently graded the model’s submitted proof, with scores finalized after unanimous consensus. The model earned 35/42 points in total, enough for gold!

HUGE congratulations to the team—Sheryl Hsu, Noam Brown, and the many giants whose shoulders we stood on—for turning this crazy dream into reality! I am lucky I get to spend late nights and early mornings working alongside the very best.

Btw, we are releasing GPT-5 soon, and we’re excited for you to try it. But just to be clear: the IMO gold LLM is an experimental research model. We don’t plan to release anything with this level of math capability for several months.

(Normally I would just link to the tweet, but in this case Alexander built a thread... and Twitter threads no longer work for linking as they're only visible to users with an active Twitter account.)

Here's Wikipedia on the International Mathematical Olympiad:

It is widely regarded as the most prestigious mathematical competition in the world. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries participate. Each country sends a team of up to six students, plus one team leader, one deputy leader, and observers.

This year's event is in Sunshine Coast, Australia. Here's the web page for the event, which includes a button you can click to access a PDF of the six questions - maybe they don't link to that document directly to discourage it from being indexed.

The first of the six questions looks like this:

Alexander shared the proofs produced by the model on GitHub. They're in a slightly strange format - not quite MathML embedded in Markdown - which Alexander excuses since "it is very much an experimental model".

The most notable thing about this is that the unnamed model achieved this score without using any tools. OpenAI's Sebastien Bubeck emphasizes that here:

Just to spell it out as clearly as possible: a next-word prediction machine (because that's really what it is here, no tools no nothing) just produced genuinely creative proofs for hard, novel math problems at a level reached only by an elite handful of pre‑college prodigies.

There's a bunch more useful context in this thread by Noam Brown, including a note that this model wasn't trained specifically for IMO problems:

Typically for these AI results, like in Go/Dota/Poker/Diplomacy, researchers spend years making an AI that masters one narrow domain and does little else. But this isn’t an IMO-specific model. It’s a reasoning LLM that incorporates new experimental general-purpose techniques.

So what’s different? We developed new techniques that make LLMs a lot better at hard-to-verify tasks. IMO problems were the perfect challenge for this: proofs are pages long and take experts hours to grade. Compare that to AIME, where answers are simply an integer from 0 to 999.

Also this model thinks for a long time. o1 thought for seconds. Deep Research for minutes. This one thinks for hours. Importantly, it’s also more efficient with its thinking. And there’s a lot of room to push the test-time compute and efficiency further.

It’s worth reflecting on just how fast AI progress has been, especially in math. In 2024, AI labs were using grade school math (GSM8K) as an eval in their model releases. Since then, we’ve saturated the (high school) MATH benchmark, then AIME, and now are at IMO gold. [...]

When you work at a frontier lab, you usually know where frontier capabilities are months before anyone else. But this result is brand new, using recently developed techniques. It was a surprise even to many researchers at OpenAI. Today, everyone gets to see where the frontier is.

# 19th July 2025, 4:27 pm / mathematics, ai, openai, generative-ai, llms, llm-reasoning

Largest known prime number (via) Discovered on 12th October 2024 by the Great Internet Mersenne Prime Search. The new largest prime number is 2^136279841-1 - 41,024,320 digits long.

# 2nd January 2025, 7:39 am / mathematics

An animated introduction to Fourier Series (via) Outstanding essay and collection of animated explanations (created using p5.js) by Andrei Ciobanu explaining Fourier transforms, starting with circles, pi, radians and building up from there.

I found Fourier stuff only really clicked for me when it was accompanied by clear animated visuals, and these are a beautiful example of those done really well.

# 5th June 2024, 3:43 pm / mathematics, processing, explorables

Google DeepMind used a large language model to solve an unsolvable math problem. I’d been wondering how long it would be before we saw this happen: a genuine new scientific discovery found with the aid of a Large Language Model.

DeepMind found a solution to the previously open “cap set” problem using Codey, a fine-tuned variant of PaLM 2 specializing in code. They used it to generate Python code and found a solution after “a couple of million suggestions and a few dozen repetitions of the overall process”.

# 16th December 2023, 1:37 am / google, mathematics, ai, generative-ai, llms

An Interactive Introduction to Fourier Transforms (via) I love interactive exploitable explanations and this is the best I’ve seen in a while: Jez Swanson breaks down exactly what a Fourier transform does, first by letting you interactively draw and deconstruct wave patterns and then by showing Epicycles andcexplsining JPEG compression. All with not a formula in sight!

# 12th January 2019, 2:55 am / mathematics, explorables

Google Image Charts: Mathematical (TeX) Formulas (via) I’m not sure when they added this, but you can now use the Google Charts Image API to render mathematical formulas, specified using TeX syntax. Wordpress.com and Wikipedia have both offered this feature for quite a while, but now you can use it anywhere on the Web.

# 12th February 2010, 9:42 am / formula, google, google-charts, mathematics, tex

GPS and Relativity (via) GPS satellite clock ticks need an accuracy of 20-30 nanoseconds. The satellites move fast enough that their clocks fall behind by 7 microseconds a day due to time dilation, but orbit high enough that the curvature of spacetime due to the Earth’s mass puts them forward by another 45 microseconds. GPS receivers have to perform relativistic calculations to determine their location!

# 11th January 2010, 9:17 am / gps, mathematics, physics, relativity, spacescience

Mobius Sliced Linked Bagel. “It is much more fun to put cream cheese on these bagels than on an ordinary bagel. In additional to the intellectual stimulation, you get more cream cheese, because there is slightly more surface area.”

# 9th December 2009, 8:03 am / bagels, breakfast, food, funny, mathematics, mobius

Pythonic Geometry. Teaching maths and Python to 13 year olds

# 15th December 2003, 2:28 am / mathematics