By now I've gotten countless emails from students studying for math olympiads that amounts to "okay, but what do I do?". So here is, at long last, a list of suggestions.
Readers may notice this is actually just an abridged version of the longer recommendations page on my site. It turns out if you make a list shorter, people are more likely to pick up the pencil and start cracking.
I really want to stress these are mere suggestions. Just because you have done everything on this list does not mean you will achieve your goals. Conversely, there are many fantastic resources that are not included on this list, since I wanted to keep this list very short (and also due to my own ignorance). If you are looking for a list of materials which are "guaranteed to be enough" for solving IMO #1 and #4, then unfortunately I can't help you.
See also the math contest FAQ's for some more philosophical (and less concrete) advice on studying.
- Algebra, combo, NT: OTIS Excerpts (by me)
- Combinatorics: Pranav A. Sriram
- Geometry: E.G.M.O. (by me)
If you've finished these and want more, I have a bunch more olympiad handouts and links to other sources. (The Excerpts contain adaptations from many of the handouts, but there are some differences, too.)
A mandatory part of any olympiad training is doing lots of problems from past contests. (I used to carry a binder with printouts of the IMO shortlist and check them off as I figured them out.)
Here are some places to start (roughly ascending order of difficulty):
The bottom of the recommendations page has some more suggestions for problems if this list isn't sufficient.