Olympiad General Preparation:
1. Math Olympiad Dark Arts
2. Methods of Solving Nonstandard Problems - Grigorieva (2015)
The classical geometry resources are still the superior choices for study, even though they are very dense. Start with 1 and 2 (CPIG and Greitzer), but everything you will need can be found in Altshiller-Court,Johnson, or Aref.
Classical Olympiad Study:
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 Olympiad Level Presentations:
6. Methods of Solving Complex Geometry Problems by Ellina Grigorieva (Springer, 2013).pdf"
7. Problem-Solving and Selected Topics in Euclidean Geometry In the Spirit of the Mathematical Olympiads by Louridas, Rassias (2013).pdf
Algebra: Inequalities - (Geometric and Analytic)
The modern resources are far superior choices for study. Start with the tutorials and then the books, everything you will need is there. 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.
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 Olympiad Level Presentations:
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 Olympiad Level Study:
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 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 resources on olympiad functional equations problems. It was all hit or miss 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, Graph Theory):
The modern treatments are far superior to the classical resources. There are a number of good textbooks for background, but often include too much. The Art of Problem Solving Intermediate Counting is a good book to start with.
1. Counting, 2nd Edition - Meng, Guan (2013)
2. Principles and Techniques in Combinatorics - Chen Chuan-Chong, Koh Khee-Meng (WS, 1992).pdf
3. Problem Solving Methods in Combinatorics - An Approach to Olympiad Problems - Pablo Soberon Bravo (2013).pdf
The background for number theory can be found in any of dozens books that are usually titled "elementary number theory" or some variation. 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. 1001 Problems in Classical Number Theory (Problems).pdf
3. 1001 Problems in Classical Number Theory (Solutions).pdf
4. 250 Problems in Elementary Number Theory - Sierpinski (1970).pdf
5. An Introduction to Diophantine Equations - A Problem-Based Approach - Andreescu, Andrica and Cucurezeanu (Birk, 2011).pdf
Χαράλαμπος Κ. Φιλιππίδης