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jsteinhardt comments on Training for math olympiads - Less Wrong
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(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.
This is a reasonable portion of what I did for math olympiads; the other parts were doing lots and lots of problems and acquiring a solid technical background.
One thing I worked on in particular is formulating good solution strategies, where I could see the general steps of a solution (or the steps of a good approach) without having to actually fill in all the details of the approach; this involves having good heuristics for figuring out what is true/false and what can be proved without too much effort (and then deferring the actual proof until later when all the pieces have come into place).
FTR I was a USAMO winner (top 12) in high school but didn't make it to the IMO team. I did make the IOI team though. I'm currently coaching at the US IOI training camp for the next week, so I don't have much spare time right now, but maybe when I have some I'll share in a bit more detail the things I did (just a warning, this was before I started optimizing my behavior so while it apparently worked it was probably not an optimal trajectory).
(I expect top programming contests are harder/less feasible to train for at levels outside raw ability. It all happens too fast.)