In terms of book recommendations, if you're interested, I highly recommend "Problem Solving Strategies" by Arthur Engel. In addition to being useful in general, it divides olympiad problem strategies in what I believe are natural categories; you could work on teaching yourself to recognize, by looking at a problem, which broad types of strategies are likely to work for it, and get yourself closer to a solution that way.
It also helps to look at all past problems in a specific area of math, and see the specific results and theorems that they use. Then you can approach all problems in the area with those tools in mind. This has helped me a great deal when preparing for the Putnam (I used it for linear algebra). It will be more helpful with some areas than others (for geometry more than combinatorics, for instance).
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:
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,