Problem solving and proofs at the Olympiad level are an entirely different skill from the AMC and AIME competitions.

There are a number of books both classical and modern the cover non-routine problem solving at the Olympiad level.

The classical resources on problem solving are mostly by the famous mathematician George Polya.

The famous general collections from Russia and Poland are classic and should be well studied.

The AopS books Art of Problem Solving volumes 1 and 2 are also well recommended.

Advanced Modern treatments:: |

1. Math Olympiad Dark Arts |

**Proof Techniques:**

**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 (CPIG and Geometry Revisited), however to do well on the Olympiad, you will need to study 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 plane geometry success is right here.

Modern treatments:: |

6. Problem-Solving and Selected Topics in Euclidean Geometry In the Spirit of the Mathematical Olympiads by Louridas, Rassias (2013).pdf |

**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(also extensive Plane and Solid Geometry sections) |

2. Problems in Higher Algebra - Faddeev |

3. A Problem Book in Algebra - Krechmar |

**Algebra: Inequalities - (Geometric and Analytic)**

The modern resources are far superior choices for study than the older books.

Start with the tutorials and then the modern and then if your really inspired take a look at the classical 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:**

**Algebra: (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 books, then it's just a matter of doing problems. Treat each one as a puzzle.

**Discrete Mathematics (Combinatorics 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.

The Art of Problem Solving Intermediate Counting is good also.

**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.

Note that at the olympiad level, you now must also know quadratic reciprocity.
The ones I like are by Roberts, and by Dudley. Ther Roberts book is very unusual for style.

Once you know the basics it really is all about doing problems.