Olympiad Problem Solving:

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.

Classical treatments and General Olympiad Problem Solving Books:
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
5. Mathematical Problems and Puzzles from the Polish Mathematical Olympiads - Straszewicz (1965)
6. USSR Olympiad Problem Book (The) - Shklasrsky, Chentzov, and Yaglom (1993, Dover) (1-1)

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.

Classical treatments:
1. Challenging Problems in Geometry by Alfred Posamentier.pdf
2. Geometry Revisited (New Mathematical Library 19) by H. Coxeter, S. Greitzer (MSA, 1967).pdf
3. An Introduction to the Modern Geometry of the Triangle and the Circle by Nathan Altshiller-Court (Dover 2007).pdf"
4. Advanced Euclidean Geometry by Roger Johnson (Dover, 1960).pdf
5. Problems and Solutions in Euclidean Geometry by Aref, Wernick (Dover, 1968).pdf"
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:
1. A less than B (Inequalities) - Kedlaya (1999).pdf (37 page introduction)
2. Topics in Inequalities 1st edition - Hojoo Lee (2007).pdf (82 pages)
3. Olympiad Inequalities - Thomas Mildorf (2006).pdf (the basic 12)
Modern Treatments:
4. Inequalities A Mathematical Olympiad Approach - Manfrino, Ortega, and Delgado (Birkhauser, 2009).pdf
5. Basics of Olympiad Inequalities - Riasat S.(2008).pdf
6. Inequalities - Theorems, Techniques, and Selected Problems - Cvetkovski (Springer, 2011).pdf
7. Equations and Inequalities - Elementary Problems and Theorems in Algebra and Number Theory - Jiri Herman (2000, CMS).pdf (Chapter 2)
Classical Treatments:
Elementary Inequalities - Mitrinovic, et. al. (1964, Noordhoff).pdf
Geometric Inequalities - Bottema, et. al. (1968).pdf
An Introduction To Inequalities (New Mathematical Library 3) - Beckenbach and Bellman.pdf
Geometric Inequalities (New Mathematical Library 4) - Kazarinoff.pdf
Analytic Inequalities - Kazarinoff (1961, Holt).pdf
Analytic Inequalities - Mitrinovic, Dragoslav S., (Springer, 1970).pdf
Inequalities - Beckenbach E., Bellman R. 1961.pdf
Additional Olympiad Inequalities Problem Books and Reference:
Algebraic Inequalities (Old and New Methods) - Cirtoaje.pdf
Old and New Inequalities - Andreescu.pdf
Secrets in Inequalities (volume 1) Pham Kim Hung.pdf
Geometric Problems on Maxima and Minima - Titu Andreescu, Oleg Mushkarov, Luchezar Stoyanov.pdf
An Introduction To The Art of Mathematical Inequalities - Steele, J. Michael (2004, MAA).pdf
When Less is More - Visualizing Basic Inequalities (Dolciani 36) - Alsina and Nelson (2009, MAA).pdf

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.

1. The Quest for Functions (Tutorial - Beginner) by Vaderlind (2005).
2. Functional Equations (Tutorial - Advanced) by Radovanovic (2007).
3. Functional Equations by Andreescu, Boreico (2007)
4. Functional Equations and How To Solve Them by Small (Springer, 2007)
5. Functional Equations by Leigh-Lancaster (2006).
6. 100 Functional Equations from AoPS.

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.

1. Applied Combinatorics by Alan Tucker
2. Counting, 2nd Edition - Meng, Guan (2013)
3. Principles and Techniques in Combinatorics - Chen Chuan-Chong, Koh Khee-Meng (WS, 1992).pdf
4. Combinatorics - Vilenkin N.(1971).pdf

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.

1. Elementary Number Theory - A Problem Solving Approach - Roberts (MIT, 1977).pdf
2. Elementary Number Theory - Dudley
3. 250 Problems in Elementary Number Theory - Sierpinski (1970).pdf
4. An Introduction to Diophantine Equations - A Problem-Based Approach - Andreescu, Andrica and Cucurezeanu (Birk, 2011).pdf
5. 1001 Problems in Classical Number Theory (Problems).pdf