Lately I've resolved to try harder at teaching myself math so I have a better shot at the international olympiad (IMO). These basically involve getting, say, three really hard math problems and trying your best to solve them within 5 hours.
My current state:
- I have worked through a general math problem-solving guide (Art and Craft of Problem-Solving), a general math olympiad guide (A Primer for Mathematics Competitions) and practice problems.
- I've added all problems and solutions and theorems and techniques into an Anki deck. When reviewing, I do not re-solve the problem, I only try to remember any key insights and outline the solution method.
- I am doing n-back, ~20 sessions (1 hour) daily, in an attempt to increase my general intelligence (my IQ is ~125, sd 15).
- I am working almost permanently; akrasia is not much of a problem.
- I am not _yet_ at the level of IMO medallists.
What does the intrumental-rationality skill of LWers have to say about this? What recommendations do you guys have for improving problem-solving ability, in general and specifically for olympiad-type environments? Specifically,
- How should I spread my time between n-backing, solving problems, and learning more potentially-useful math?
- Should I take any nootropics? I am currently looking to procure some fish oil (I don't consume any normally) and perhaps a racetam. I have been experimenting with cycling caffeine weekends on, weekdays off (to prevent tolerance being developed), with moderate success (Monday withdrawal really sucks, but Saturday is awesome).
- Should I add the problems to Anki? It takes time to create the cards and review them; is that time better spent doing more problems?
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).