Training for math olympiads

(Take with a grain of salt, I'm far from IMO level and never seriously trained for math contests.)

After solving any given problem, reflect on general methods that would allow solving a bigger class of problems including the one you've cracked. For any miracle of intuitively seeing a solution method (or noticing some useful property), look for ways of more systematically inferring a workable method that don't rely on miracles. Don't consider a problem solved just because you solved it (i.e. used your intuition), you should also figure out how it could be solved (i.e. know in more detail how your intuition figured it out, or know a method other than the unknown one used by your intuition).

I expect this can get one past some limitations of raw ability that wouldn't otherwise be lifted using just problem-solving practice, but I'm not sure how far.

Learn to convert time into reliability of a solution. Don't just solve a problem (in the contest setting), but check correctness of the solution from as many angles as remaining time allows.

Generalize the problem, solve in general, check that the general solution gives the same answer on edge cases as straightforward solutions in those cases. Solve using a different method. Infer additional facts about the problem that it doesn't ask you to infer, and infer from those facts other facts you encountered. Reduce the problem or parts of the problem to different formulations, solve in those different formulations, translate back, check that it fits. Invent redundant overlapping subproblems just to compare intermediate results. (Whichever of these is most natural.)

(This is the greatest piece of low-hanging fruit that I never collected. In particular, knowing this trick of converting time into reliability allows to get perfect scores on simpler tests and not lose points for harder problems that you know how to solve in principle when there's enough time.)

Sorry not to have answered you earlier, I have been absent for the first days in the final round of selection for my country's IMO team, which after years of practice I am finally on. Feel free to PM me if you have more questions, though I can't promise that I'll know all the answers.

As far as practice vs learning new maths goes, the IMO has something of an unwritten syllabus of theorems you need to know. If you don't know everything on it, then by far your best route is to learn the things on it. Once you've got that, learning more theorems is unlikely to help much, and you just need practice, practice and more practice. I have tried to solve at least one problem every day for the past year.

The one exception is that if you start to find that you simply lack the raw talent required in one ore more of the areas, then try learning unconventional techniques for doing it, you may be better at those.

If you're not sure which stage you're at yet, try some problems on Art Of Problem Solving, then look at the solutions of the ones you couldn't do within a day or so. If you understand all the solutions then you probably have a sufficient knowledge base.

A trap to avoid falling into is only practising the areas you are good at, this is a very seductive mistake since inevitably those areas will feel more fun to do. Try the areas that seem hard and boring until they become fun, but if you start to find yourself disliking maths in general switch back to something you enjoy.

If possible try to find a mentor of some kind, someone you can meet in real life is best. This is especially essential if you're still at the stage where you don't know the whole unwritten syllabus yet.

This type of approach may not be the best if you actually wish to become a good mathematician, but the IMO is sufficiently competitive that if you are unlikely to get on with anything less (I only made it by the tiniest margin as it is).

Concerning nootropics, for the most part what little evidence there is for them working is measured on people that have aged or damaged brains.

Pop quiz. This was one of my favorite questions that I recall back in my middle school math team days (it's probably easy for a mathy adult though): "What is the length of a line that spirals once around the surface of a cylinder as it descends from top to bottom?"

http://markan.net/mathcontests.html

I'm not sure what level of math you're at, but doing previous IMO problems would presumably be useful; here's a book containing over 1900, with solutions.

(Btw, the authors' grasp of English is a little shakey at times, and they rewrote some of the problem statements, so be warned.)

Do math contest problems from the different areas of math covered by the IMO and start at your current level of ability and work from there.

Find problems from books, previous contests and forums like Art of Problem Solving.

In terms of book recommendations, if you're interested, I highly recommend "Problem Solving Strategies" by Arthur Engel. In addition to being useful in general, it divides olympiad problem strategies in what I believe are natural categories; you could work on teaching yourself to recognize, by looking at a problem, which broad types of strategies are likely to work for it, and get yourself closer to a solution that way.

It also helps to look at all past problems in a specific area of math, and see the specific results and theorems that they use. Then you can approach all problems in the area with those tools in mind. This has helped me a great deal when preparing for the Putnam (I used it for linear algebra). It will be more helpful with some areas than others (for geometry more than combinatorics, for instance).

• Learn the formulae for a good number of infinite sums, and for all discrete-math equations that Euler came up with.
• Remember that the most-probable answers are 0 and 1.
• Get a lot of sleep the night before the exam.