Books for Advanced algebra and Advanced geometry(AMC 12) I will be taking the American Math Contest. Various topics are covered in this test including advanced geometry and algebra. It would be great if any of you could provide books/references which contain challenging or hard problems related to advanced geometry and algebra.
 A: I am preparing for this exam as well, as I type this answer actually (A Walt S. Diophantine video is playing in the background).
I'll assume you are preparing for the AMC 10 or 12, not the 8. I'll also assume that you're familiar with the 35 official AMC exams on Art of Problem Solving. 
This website has 15 PDFs of books that have been extremely helpful, including several for advanced algebra and geometry. These contain tons of practice problems.
These Hungarian problems are also a good challenge. 
Walt S. has problem set links in the descriptions of some of his AMC youtube videos like this one. They are a great resource as well!
These should be plenty to practice from for the next two weeks. Good luck!
A: My introduction to Abstract Algebra was Fraleigh's text.  It was a very enjoyable experience.   Of course,  I had a great professor,  John Stallings. 
Lang's Algebra is also in my opinion very good (and is as far as I know quite highly regarded).
For geometry, I recommend Spivak's five volume Comprehensive Introduction to Differential Geometry.   I was happy to find my advisor Peter Petersen had a copy of this, as did I.  
(While I didn't end up getting my PhD, I did get a pass at the PhD level on the Geometry/ Topology qual at Ucla.  And the Algebra one, for that matter.)
Good luck!
A: Personally, to prepare for the AMC competitions, I do a combination of old tests and Alcumus. You can find old tests here here. Doing old tests helps you get a good feel about what to expect in the competition and allows you to quickly identify your weaknesses. That's when Alcumus comes in play. Alcumus is a resource provided by Aops. Once you create a free account, doing problems from basically all the topics except for pre-algebra will be helpful. Also, when you get a question wrong, the worked out solutions are also very helpful.
