Problem solving and proofs at the Olympiad level are an entirely different skill from the AMC and AIME competitions.
It is assumed you've completed the Art of Problem Solving Volume 1 and at least some of Volume 2. There are a number of books both classical and modern that cover non-routine problem solving at the Olympiad level. The classical resources on problem solving are mostly by the famous mathematician George Polya.
| Classical treatments: |
| 1. How to Solve It - Polya |
| 2. Mathematical Discovery Polya |
| 3. Mathematics and Plausible Reasoning I Polya |
| 4. Mathematics and Plausible Reasoning II (2nd edition) Polya |
| Modern treatments:: |
| 1. Math Olympiad Dark Arts |
Olympiad Problem Collections:
General advanced problem collections are a good place to start, covering a range of topics. They will also help you with your AIME performance, necessary for Olympiad qualification. Of course, your main focus should be to practice completely the past problems of the Olympiad you are preparing for, USAJMO, USAMO, IMO etc.
The famous general collections from Russia, Poland, and Hungary should be well studied. I find the Polish book to still be the most relevant for really learning Olympiad level proofs..
| Classical Problem Collections:: |
| 1. Problems in Elementary Mathematics - Lidsky |
| 2. Mathematical Problems and Puzzles from the Polish Mathematical Olympiads - Straszewicz |
| 3. USSR Olympiad Problem Book (The) - Shklasrsky, Chentzov, and Yaglom |
| 4. Hungarian Problem Book I (1894 - 1905) - Rapaport (MAA,1963) |
| 5. Hungarian Problem Book II (1906 - 1928) - Rapaport (MAA,1963) |
| 6. Hungarian Problem Book III (1929 - 1943) - Andy Liu (MAA,2001) |
| 7. Hungarian Problem Book IV (1947 - 1963) - Barrington, Liu, (MAA,2011) |
Geometry: Plane Geometry
It is assumed you've completed the Art of Problem Solving Introduction to Geometry.
In my view, the classical plane geometry resources are still the superior choices for study, even though they are very dense.
Start with #1 and #2 (Challenging Problems and Geometry Revisited), however, to do well on the Olympiad, you will need to study the more advanced Altshiller-Court, Johnson, and Aref.
Altshiller-Court and Johnson are very light on problems, Aref is heavy on problems, so they all work together.
Everything you need for Olympiad plane geometry success is right here.
| Modern treatments:: |
| 1. Problem-Solving and Selected Topics in Euclidean Geometry In the Spirit of the Mathematical Olympiads by Louridas, Rassias (2013) |
Algebra: Equations and Trigonometry:
It is assumed that you've completed and understand both Art of Problem Solving Introduction to Algebra and Art of Problem Solving Intermediate Algebra.
In my view, the classical Algebra problem books are still the superior choices for study.
| 1. Problems in Elementary Mathematics - Lidsky |
| 2. Problems in Higher Algebra - Faddeev |
| 3. A Problem Book in Algebra - Krechmar |
Inequalities - Geometric and Analytic
The modern resources are far superior choices for study than the older books as they are oriented towards Olympiad competition study. Start with the tutorials and then the modern books and then if your really inspired take a look at the classical and other books, everything you will need is in the tutorials and modern books. The classical resources include large amounts of material that is not relevant for high school olympiad contests and though interesting, can eat up your time.
Tutorial Introductions:
Functional Equations:
There are no classical books and resources on olympiad functional equations problems. It was all hit or miss back then from various magazine problem sections. Start with the tutorials, then on to the modern books, then it's just a matter of doing problems. Treat each one as a puzzle.
Discrete Mathematics (Combinatorics, Probability, and Graph Theory):
It is assumed you've finished the Art of Problem Solving Counting and Probability book. The modern treatments are far superior to the classical resources. There are a number of good textbooks for background, but most include too much as they are oriented towards college courses. The idea is to pick one and learn it well. I always liked the Tucker book, now in a 6th edition. The Tucker and Vilenkin books have great coverage of generating functions.
| 1. Applied Combinatorics by Alan Tucker |
| 2. Counting, 2nd Edition - Meng, Guan |
| 3. Principles and Techniques in Combinatorics - Chen Chuan-Chong, Koh Khee-Meng |
| 4. Combinatorics - Vilenkin N. |
| 5. Intermediate Counting and Probability - Art of Problem Solving |
Number Theory:
It is assumed that you've covered the matieral in the Art of Problem Solving Introduction to Number Theory. The necessary background for Olympiad level number theory can be found in any of dozens of books available that are usually titled "Elementary Number Theory" or some variation. The idea is to pick one and learn it well. Generally they don't cover diophantine equations that well, which is where the Olympiad problem books come in. The Sierpinski book is the best. Note that at the IMO level, you now must also know quadratic reciprocity. The ones I like are by Roberts, LeVeque, and Dudley. The Roberts book is very unusual for style. Once you know the basics it really is all about doing problems.
Game Theory:
Analyzing a game is now a question on almost every high level olympiad. However, there are not a lot of references specifically oriented towards solving olympiad level mathematical problems. Here are a few:
| 1. Game Theory - Part I. Impartial Combinatorial Games - Ferguson |
| 2. Impartial Games - Guy |
| 3. Mathcamp Combinatorial Game Theory Tutorial (2019) |
| 4. Fair Game by Richard K. Guy, COMAP Mathematical Exploration Series, 1989 |
| 5. On Numbers and Games by J. H. Conway, Academic Press, 1976 |
| 6. Winning Ways for your mathematical plays by Berlekamp, Conway and Guy, Academic Press, 1982 |