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I am a high school sophomore and I have decided to begin studying for high school math competitions like the AMC 12, AIME, and USAMO. For those who aren't familiar, the tests come in a sequence as ordered, with fewer people taking each round, until a U.S. team is selected to compete internationally. Previously I had relied on my natural math talent to take these tests and as a result I have missed the AIME cutoff by fractions of questions for the past two years.

I was wondering what resources are best for studying for the tests. Should I learn calculus/linear algebra? Should I study by doing practice problems or by learning whole new concepts or both? Looking at IMO sample problems(PDF), I am a long way from competing at this level; I am ready to learn, but how do I proceed? Thank you.

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up vote 5 down vote accepted

Three things:

  1. http://www.artofproblemsolving.com/

  2. New Mathematical Library books #5, 11, 12, 17, 25, 27, 29, 31, 33, 38, 40, 42 http://www.maa.org/ebooks/nml.html

  3. Work LOTS of problems from old tests!

Neither calculus nor linear algebra are needed, but if you're ready for these subjects then by all means continue your studies in math by studying these subjects (when you have the appropriate background), as life does not end when you're finished with high school math contests.

(added next day) In the time since I wrote the above (which was done very quickly, but it seems fine on rereading now), I've looked through my books at home and found 4 that merit mention.

1. Posamentier/Salkind, Challenging Problems in Algebra

http://www.amazon.com/dp/0486691489

This is a cheap Dover Publications (1996) reprint of a book originally published in two thin volumes in 1970 and reprinted as one volume in 1988. The difficulty of the problems (all of which have solutions) is slightly higher than the typical (strong or honors level) U.S. Algebra 2 course, which makes it an excellent place to start if your background isn't very strong and you're looking for easier AHSME level problems that stay within approximately U.S. high school Algebra 1 and 2 material.

2. George Polya, How to Solve It

http://www.amazon.com/dp/4871878309

This is a universally recommended book for contest math that was first published in 1945. The 2nd edition was published in 1957, and the 2nd edition was later reprinted in 1988 and 2004. In my opinion, the book is better for the cognitive strategies it lays out than for specific mathematical "tricks". However, because of this the value of the book will increase as you learn more mathematics and get more experience in contest problem solving.

3. Arthur Engel, Problem-Solving Strategies

http://www.amazon.com/dp/0387982191

This is an encyclopedic book (first published in 1998) that contains a huge number (over 1300) problems, all of which are either fully worked or provided with a substantial hint and/or solution outline. This is the most complete and extensive book on contest math problems that I am aware of.

4. Steve Olson, Count Down

http://www.amazon.com/dp/0618562125

This book was first published in 2004 and it gives a "popular" account of the 2001 U.S. IMO team and that team's IMO performance in 2001. Included are discussions, for each of the 6 IMO problems from 2001, on how various contestants solved the problems.

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I'd like to stress that hard work is usually what makes someone great at something. Michael Jordan was exceptionally gifted athletically but that didn't make him the greatest ever. His exceptional ability combined with his exceptional work ethic are what made him the greatest ever. It is well known that he worked extremely hard, more than his peers. Kobe Bryant is probably the hardest worker of all current players. I always wanted to be good at contests but I never put in any work. If you actually follow all that Dave Renfro said, or half of it, you will learn a lot and improve greatly. –  Graphth Apr 13 '12 at 14:06
    
I actually had the privilege of reading the book Count Down and got motivated by it.I guess it is indeed about work ethic. –  Eisen Apr 29 '12 at 12:24
    
In fact, I would like to criticize Engel on one front:that book will tell you how to solve some kinds of problems but mind you, the USAMO is actually the most unorthodox olympiad in the world.Those who propose the questions are aware of the techniques people have invented to solve particular type of problems, so beware! –  Eisen Apr 29 '12 at 12:26
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