Mathematical Research vs Mathematical Olympiad
Explore the differences between doing mathematical research and mathematical Olympiad problems, and the kind of profile that moves each type.
The only way of learning mathematics is doing mathematics.
Paul Halmos
High School Background
I was one of those kids who did not like maths since the very beginning. At that time, I would rather do experimental physics, specifically the astro stuff, and fall in wonder with the beauty of the cosmos around us. However, as I got to the end of uni, I started to get more involved in the art of problem solving, and naturally, maths came as a natural interest.
Right at the end of high school, I started to get involved in some mathematics problem-solving contests which had the name of math Olympiads. I was there only for the fun, but later I found out people were very keen on problem-solving contests and studied very hard for some of them internationally.
I did not do well in them, however I was always keen to know the solutions and study them — and in the end, I had fun.
My Olympiad experience
In math Olympiads, people challenge themselves in a setting where there’s a sheet with around 4 to 5 exercises that are supposed to stimulate mathematical reasoning. These exercises typically have very elegant solutions and require the participant to look at things from the right angle.
When I was participating in math Olympiads, my experience was not really around:
- How can I use differentiability to solve this problem?
- What is the underlying theory around this question?
and more about:
- What clever mathematical patterns are being used here?
- What clever mathematical tricks can I use here?
In a sense, it was the ultimate contest of problem solving. You didn’t really need to know much theory at all, besides knowing how to be clever with mathematical manipulation and some basic school mathematics facts.
Another important aspect around these questions is that they are of closed form and made to be typically solvable in a couple of hours, whereas in research, questions tend to be open-ended and take months to years on average.
University Background
After high school graduation, I started to learn mathematics from a formal standpoint, where I was presented with many wonderful theories and eventually decided to do a PhD in maths, where I was met with what it means to be a researcher in mathematics.
My Research experience
Research mathematics differs from Olympiad mathematics in the sense that it’s not really about problem solving, but more about theory building. For example, a problem in a mathematical Olympiad might be very challenging to solve and have a very beautiful solution. However, if this problem doesn’t constitute a building block or lead to a better understanding in theory building, then it’s not considered good mathematical research. This is the key difference that changes everything.
There is good mathematical research out there that is fairly simple when it comes to problem solving and more about connecting the dots. This doesn’t happen in mathematical Olympiads, where the beauty of the solution is the main focus — not the problem itself.
Another major difference is that research questions are supposed to have unknown answers, whereas Olympiad mathematics is supposed to have known, elegant solutions. (And yes, this does mean that a research question might end up being a math Olympiad question, depending on the criteria we just talked about.)
Concrete Examples of Questions
To put it in a concrete scenario, a nice question in mathematics that is terrible for mathematical Olympiads is given by:
- “Does every compact Riemannian manifold admit a metric with constant scalar curvature?”
This question is open-ended and requires a deep understanding of, in this case, differential geometry — making it suitable for research, but not for an Olympiad setting.
On the other hand, one might consider the reverse situation:
- “Prove that for any positive integer n, the sum ∑ₖ₌₁ⁿ k² = (n(n+1)(2n+1))/6.”
This question doesn’t require prior knowledge to be understood, only clever manipulation, and it has a very elegant solution.
Different People, Different Roles
Being good at mathematical Olympiads is not the same as being a good researcher — and vice versa. In fact, being a good researcher is something super niche and requires personal traits that are 95% different (I’ll explain later the other 5%) from being good at Olympiads.
A good Olympiad participant is a quick problem-solver. These types of people tend to be very clever and are highly valuable in the market, as they can provide solutions to problems with amazing speed, which is crucial in real-world settings. However, this does not mean that the person has the capacity or resilience to learn mathematical theory. In fact, I’ve met people who couldn’t be more bored with advanced mathematics — for example, struggling to grasp concepts that require hours of study and tinkering. These people are here to think, not to learn. Their dopamine hit comes from solving, not learning, which doesn’t go so well in a formal research setting, where 95% of the work is learning what others have done, and the other 5% is our little bit of thinking on what we can add to make something new.
People with not-so-great problem-solving skills but who are curious might actually thrive in a research setting, where their curiosity drives them to learn more — making the 95% of the work achievable. Then, the other 5% might come through collaboration or just some time spent thinking. However, put them in a math Olympiad frame and things go south quickly.
A major point where these characteristics are clearly visible is when it comes to developing a thesis in the final years of study. People who get their dopamine from solving — and not from learning — have a really hard time finishing these types of documents. Meanwhile, people who get their dopamine hit from the process of learning tend to overachieve in these settings.
Personal Statement
In my case, I’m way more aligned with the mathematical researcher mindset. Curiosity is my main driver — I get my dopamine from the process of learning and finding the truth, not from the challenge of solving, and to be fair I am not a quick solver either.
On a final note, I believe it’s important to point out that everything I’ve written here is nonsense if taken too seriously, and should be taken with a grain of salt. Human beings are very special creatures who don’t follow patterns. (And that’s a good thing!)
If you’d like to read more about the differences between these kinds of people, make sure you check out the blog of mathematics made by Terence Tao — he’s written plenty on this topic, and his blog posts are absolutely delicious. Check as well the subreddits r/math where this debate is fierce.
Newcastle Upon Tyne, England
Tiago Verissimo
