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.)
For the USAMO/IMO, the ability to solve a single problem is already a large step. Solving 4 out of the 6 problems means you almost surely make the IMO team. I think the issue is more how he can get good enough to solve such difficult problems (they are also proofbased, so an actual solution should always get full points barring a bad writeup).
One core skill is ability to solve problems eventually, to not stop before you solved the problem. Bringing this to contest setting requires speeding up to a level that is probably unattainable because of limitations of raw ability, and might benefit from lots of training. But another failure mode is stopping right after you solve the problem, so that you won’t notice a stupid error, even though you’ve got the solution. This is a piece of low-hanging fruit that doesn’t necessarily require any significant effort (in comparison), but still needs to be collected.
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.
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.)
For the USAMO/IMO, the ability to solve a single problem is already a large step. Solving 4 out of the 6 problems means you almost surely make the IMO team. I think the issue is more how he can get good enough to solve such difficult problems (they are also proofbased, so an actual solution should always get full points barring a bad writeup).
One core skill is ability to solve problems eventually, to not stop before you solved the problem. Bringing this to contest setting requires speeding up to a level that is probably unattainable because of limitations of raw ability, and might benefit from lots of training. But another failure mode is stopping right after you solve the problem, so that you won’t notice a stupid error, even though you’ve got the solution. This is a piece of low-hanging fruit that doesn’t necessarily require any significant effort (in comparison), but still needs to be collected.
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”.