PhilGoetz comments on Training for math olympiads - Less Wrong
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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.)
The math olympiad is not like most tests - each problem has one or more key insights you must have in order to solve it. Get the insight; solve the problem. Don't have the insight, don't solve the problem.
This piece of advice was for preventing "get the insight, solve the problem incorrectly by making a stupid technical error".