Why Competitive Math Matters More Than Ever in the Age of AI
There's something magical about that moment when you're staring at a math problem that seems impossible, and then suddenly—click—you see it. The solution becomes crystal clear, and you wonder how you ever missed something so elegant. That's the essence of competitive mathematics, and it might just be the most important type of thinking we can develop in our AI-dominated world.
Let me tell you why.
What Makes Competitive Math Different
Most people think of math competitions as just harder versions of regular math class. But that misses the point entirely. The difference isn't just difficulty—it's a completely different way of thinking.
In your typical high school algebra class, you learn what a polynomial is, how to find its roots, maybe even Vieta's formulas if you're lucky. Then on the test, you get problems that clearly signal which technique to use. It's pattern recognition dressed up as problem-solving.
Competitive math throws that playbook out the window.
Here's a simple example that illustrates the difference: Prove that every positive integer greater than 11 can be written as the sum of two composite numbers.
Most high schoolers would stare at this and have no idea where to start. There's no obvious formula to apply, no clear category of problem to recognize. But there's an elegant solution hiding in plain sight.
If the number is even, write it as 2k. Then express it as (2k - 8) + 8. The first term is even (so composite), and 8 is obviously composite. If the number is odd, write it as 2k + 1, then express it as (2k - 8) + 9. Again, the first term is even and composite, while 9 is composite.
That's it. Simple, clean, and completely non-obvious until you see it.
This is what competitive math is really about: taking familiar concepts and applying them in ways that require genuine insight, not just pattern matching.
The Aha Moment
Po-Shen Loh, who coaches the US Math Olympiad team, has a brilliant way of describing why the Putnam Competition—one of the hardest math contests in the world—is so difficult. The Putnam has twelve problems worth 10 points each. The median score is zero. The mean is around 1.
Problem 1 requires one "aha moment"—you might try 100 different approaches, and one will work. Problem 2 needs two aha moments in sequence. By the time you get to Problem 6, you're looking at a decision tree with depth six, where each branch represents a potential insight that might or might not lead somewhere useful.
Think about that for a moment. You need to have the right insight, then another right insight building on the first, then another, and so on, six layers deep. It's like navigating a maze where you can't see more than one step ahead, and most paths lead nowhere.
This isn't the kind of thinking you develop by memorizing formulas or following worked examples. It's mathematical intuition in its purest form.
Why This Matters Now More Than Ever
Here's where things get interesting—and a bit concerning.
AI is phenomenal at the kind of mathematics we typically teach. Pattern recognition? Formulas and calculations? Applying known techniques to familiar problems? AI crushes humans at all of this. GPT-4 can probably solve most problems from your high school or even undergraduate math courses because they follow predictable patterns.
But something remarkable happened recently: OpenAI announced that their internal reasoning model achieved gold medal performance on this year's International Mathematical Olympiad. An AI system solved problems that require genuine mathematical creativity—the kind where you need to see patterns that aren't immediately obvious, where you need to combine ideas in novel ways.
This should be a wake-up call. If AI can now handle even creative mathematical problem-solving, what does that mean for human mathematicians? For students? For anyone who works with quantitative reasoning?
I think it means we need to double down on developing mathematical maturity—the kind of deep, flexible thinking that competitive math cultivates.
What Mathematical Maturity Really Means
Mathematical maturity isn't about knowing more formulas or being faster at calculations. It's about how you approach problems when you don't immediately know what to do.
It's the ability to look at a problem like this year's IMO Problem 6—which asked about the minimum number of rectangular tiles needed to cover an n×n grid such that exactly one tile in each row and column remains uncovered—and recognize that you've seen similar patterns in museum floors or hotel lobbies. It's having the intuition to think: "This feels like a tiling problem I've encountered before."
Most undergraduate courses, unfortunately, focus on what I call "plug-and-chug" mathematics. You learn a technique, practice it on similar problems, then apply it on exams. There's value in this, but it's exactly the kind of thinking that AI can replicate easily.
Mathematical maturity is different. It's about:
Looking at unfamiliar problems and developing strategies from scratch
Recognizing when a problem might benefit from an unexpected approach
Combining techniques in novel ways
Having the persistence to work through problems that don't have obvious solutions
This is what competitive math trains, and it's what we desperately need more of.
The Beauty of Mathematical Thinking
Let me share one more example that captures why I find competitive math so compelling.
In competition problems involving polynomials with roots a, b, and c, you might be asked to find the value of a² + b² + c² or a³ + b³ + c³. Students who've never seen Olympiad problems would try to solve for a, b, and c individually—a nearly impossible task for most competition problems.
But there's a more elegant approach. For a² + b² + c², you can use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Vieta's formulas give you a + b + c and ab + bc + ca directly from the polynomial's coefficients, so you can solve for a² + b² + c² without ever finding the individual roots.
For a³ + b³ + c³, you can use the identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), which again lets you work with symmetric sums rather than individual values.
These aren't just tricks—they're glimpses into the deeper structure of mathematics. They show how the right perspective can transform an impossible problem into an elegant solution.
Where We Go From Here
I'm not suggesting that everyone should become a math competition specialist. But I do think everyone in STEM fields should experience this type of mathematical thinking, especially as AI capabilities continue to advance.
We need people who can look at problems with fresh eyes, who can combine ideas in unexpected ways, who can push beyond the obvious approaches when the standard methods don't work. These are fundamentally human skills—the kinds of things that, even as AI gets more powerful, will remain valuable precisely because they're so hard to systematize.
The challenge is figuring out how to incorporate this kind of thinking into regular mathematics education. You can't just make every course as hard as the Putnam—everyone would fail, and that wouldn't help anyone. But we can find ways to include more open-ended problems, more opportunities for genuine discovery, more moments where students have to think creatively rather than just apply memorized procedures.
Maybe it's adding contest-style problems to regular courses. Maybe it's creating space for mathematical exploration and conjecture. Maybe it's simply asking "why" more often, and accepting that sometimes the answer is "let's figure it out together."
What I know is this: the aha moment you get from solving a truly challenging problem—that flash of insight when everything clicks into place—is one of the most satisfying experiences in mathematics. It's also training for the kind of thinking our AI-augmented future will demand.
The question isn't whether competitive math problems are hard. They are. The question is whether we're preparing people for a world where the easy problems get solved by machines, and the interesting work requires the kind of mathematical maturity that only comes from wrestling with problems that don't have obvious solutions.
I think we are, one aha moment at a time.
