The AI that solved IMO Geometry Problems | Guest video by @Aleph0

In January 2024, Google DeepMind released an AI model called Alpha Geometry, which could solve geometry problems from the International Mathematical Olympiad or the IMO. The IMO is the highest level of competitive math contest at the high school level. Every year, more than 100 countries send six teenagers to represent them at the competition. Each country has its own elaborate system of contests leading to their choice of six representatives.

When alpha geometry was tested on a database of 30 geometry problems, it solved 25 of them. This is better than the performance of a silver medalist. This quite understandably generated a lot of buzz. But here's what nobody talks about.

The most surprising part for me is not that AI managed to solve these problems, but it's what happened even before the AI showed up. A 25-year-old technique with no AI at all, just using logic and pure equation solving, was already able to solve 18 out of 25 problems. That's already a bronze medal at the IMO. And then the model hit a wall.

There were problems that even this ingenious logical model wasn't able to solve. And that's where AI comes in. But instead of leaving everything to AI, the DeepMind team very cleverly integrated the logical model with an AI component in order to reach their spectacular result of 25 out of 30 problems. So in this video, yes, I do want to talk about AI and geometry.

But before that, I also want to talk about how a nonAI model was already better at geometry than the vast majority of humans. So with that, let's begin. Our task today is simple. Build a bot that can solve IMO geometry problems.

But I'm going to make a caveat to make our problem a little bit harder. Write a bot that solves IMO problems with no AI. To begin attacking this, we're going to use a well-kept secret about geometry problems. A few key facts can get you very far.

For example, here are two important facts from geometry. One, when two lines cross, the opposite angles are equal. And two, if these two horizontal lines are parallel and you look at the Z over here in blue, the angles on the inside of the Z are equal. With these two facts alone, we can already prove a non-trivial theorem.

In this diagram, the two orange ...