What books would you guys recommend to learn number theory, geometry, combinatorics, and algebra at a level appropriate for the AIME and/or USAMO? With the month of February approaching, the month the AMC (American Mathematics Competition) series takes place in, I'm realizing how much more I have left to learn.
Thanks
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I find Larson "Problem Solving Through Problems" a great book. It's problem after problem organized by approaches and topics. Many of the solutions are particularly clever. Yet he breaks the thought process into steps so you can really gain an insight as to how to solve problems that look pretty challenging at first glance.
http://www.amazon.com/Problem-Solving-Through-Problems-Problem-Mathematics/dp/0387961712
As Sameer mentioned in a comment, The Art and Craft of Problem Solving by Paul Zeitz is a frequently cited book for this kind of contest preparation.
However, I think the most efficient way to get better at doing these contest problems is to buckle down and solve hundreds of them. AoPS has a huge collection of problems that will definitely help. If you find that you can't solve a problem you can click on the problem and there will usually be a forum thread on it, and hopefully someone will have answered it and hopefully the solution will be lucid enough to be helpful. If not, you can always ask here.
EDIT: link to USA-specific contest problems (including AIME/USAMO)