Some 3am ramblings for my IMO guest-session students (back in 2019).
Well, it’s 3am, and I have a shitload of work to do, but sometimes inspiration decides to hit at weird times.
I think I had better write this given all the sessions I’m about to do.
The short answer is: because it goes against the idea of olympiads.
The whole point of Olympids is that you have a “hard” problem that is somehow solvable with elementary techniques - things that you don’t need to know beforehand. This is partly due to equity reasons, which is also why stuff like calculus and group theory isn’t tested on olympiads - these aren’t part of standard high school curricula. (3-variable inequalities though, right?) Also for secondary/JC kids, there’s a tendency to be obsessed with the theory and jargon, forgetting all about the underlying idea.
I think there’s also quite a basic difference in learning styles: for Olympiad training much is focused on problem solving. The trainings are problem sets. The tests are problem sets. However, if you ever try to learn something new in math, it is usually presented in a way that goes like “here’s 5 definitions, then a couple of basic propositions, and then a bunch of theorems” etc. It’s annoying at first, because you never know what various things are used for (and why we care), then immediately there’s an attempt to reason complicated things using the notation you’ve read less than a minute ago, and your brain starts to overclock just trying to comprehend the theorem statements. Here is roughly where you give up, go back to solving problems and maybe try again in a few months (or even never).
(Depressing Fact. This seems to be how most math in college is taught.)
(On a separate note, this is why people like you guys seem to think that college is hard. But this is really a matter of timescales: in Olympiads we kinda attempt to learn something new in a matter of hours, but a topic in undergraduate math can easily take a whole semester. So skimming past the first 20 definitions and theorems in 5 minutes isn’t going to be easy, yeah?)
And of course, there’s the argument from utility: the proportion of problems where theory gives you a significant advantage is very slim (just by design), so you won’t really get better at doing contests just by knowing a lot of theory (not directly, at any rate).
I think first we have to recognize what’s so good about Olympiads (beyond the shiny medals and bragging rights). What Math Olympiads train you to do is to be able to solve problems working with only the knowledge in your head. This is in comparison to, say, solving problems with the help of Google. Or coming up with new theories. Or figuring out what’s an interesting problem to think about. It’s intangibly useful, but still limited.
It really does make sense that if this is the goal, then you shouldn’t really think about theory. However, sometimes the way to problem solving is a deeper understanding, and occasionally that’s much clearer with theory. Anyone who has struggled with geometry problems and finally learnt projective theory will understand immediately: understanding cross-ratios would have finally gave structure to aimless sine chasing. The optimal level of understanding is just sometimes not attainable without the correct notations and definitions set up.
Here’s another argument from your future self. Take it from me that most things you’ve learnt for MO will be completely useless once you get to college. Except, of course, the little bit of field theory that you picked up. Maybe you spent a week figuring out the Probabilistic Method. Maybe some epsilon-delta things give you a head start in analysis. Maybe you know the notions of dimensions (in linear algebra) so well from the combinatorial applications.
I’m personally not a fan of learning theory as it is. To me, I don’t think I have the attention span to sit down with a group theory textbook just to slug through it. So the best way forward is, perhaps, backwards. Start with the problem: a problem you know perhaps uses some advanced tool, but is still simple enough for you to care about. Learn what you need to understand this particular application, but don’t be afraid to give up and try again some later time.
Anyway, for this to be even remotely useful, you have to basically understand it well enough to re-invent it yourself, because the sort of advantage it gives is kind of like recreating the problem creator’s thought process. Sometimes a problem comes from obvious facts in higher mathematics, and you’re trying to get at that. This is hardly the most efficient way to get good, but it can be pretty fun.
Also keep in mind that it’s also incredibly inefficient to learn theory by yourself. I found that personally it’s much easier when I’m thrown into the deep end in college (where my grades depend on learning the stuff). If you’ve been collecting little observations and trying to make sense of them, you’ll pick these up with almost no time and effort.