"It is better to give than to receive - especially advice" - Mark Twain

The primary study material for AIME are the advanced AoPS series texts:

If you have qualified for AIME, you have probably studied the more elementary AopS books fully, and can expect to get 4-6 or less with just these basics.
If you want to score 7+ on the AIME, you will need to study the more advanced AoPS books, as well as past AIME problems and solutions.

Past AIME Problem and Solution Sets
ProblemsSizeSolutionsSize
AIME Problems (1983-2011) 17.9 MB AIME Solutions (1983-2011) 27.5 MB
AIME I Problems (2012) 0.5 MB AIME I Solutions (2012) 0.5 MB
AIME II Problems (2012) 0.5 MB AIME II Solutions (2012) 0.5 MB
AIME I Problems (2013) 0.5 MB AIME I Solutions (2013) 0.5 MB
AIME II Problems (2013) 0.5 MB AIME II Solutions (2013) 0.5 MB
AIME I Problems (2014) 0.5 MB AIME I Solutions (2014) 0.5 MB
AIME II Problems (2014) 0.5 MB AIME II Solutions (2014) 0.5 MB
AIME I Problems (2015) 0.5 MB AIME I Solutions (2015) 0.5 MB
AIME II Problems (2015) 0.5 MB AIME II Solutions (2015) 0.5 MB
AIME I Problems (2016) 0.5 MB AIME I Solutions (2016) 0.5 MB
AIME II Problems (2016) 0.5 MB AIME II Solutions (2016) 0.5 MB
AIME I Problems (2017) 0.5 MB AIME I Solutions (2017) 0.5 MB
AIME II Problems (2017) 0.5 MB AIME II Solutions (2017) 0.5 MB
AIME I Problems (2018) 0.5 MB AIME I Solutions (2018) 0.5 MB
AIME II Problems (2018) 0.5 MB AIME II Solutions (2018) 0.5 MB
AIME I Problems (2019) 0.5 MB AIME I Solutions (2019) 0.5 MB
AIME II Problems (2019) 0.5 MB AIME II Solutions (2019) 0.5 MB

Core Study and Review Books are the AoPS Series:
AIME 4-6: (Basic AoPS Books)AIME 7+: (Advanced AoPS Books)
  • Introduction to Counting and Probability
  • Introduction to Number Theory
  • Introduction to Algebra
  • Introduction to Geometry
  • Art of Problem Solving Volume 1
  • Intermediate Counting and Probability
  • Intermediate Algebra
  • Precalculus
  • Art of Problem Solving Volume 2

AIME 7+ Topics (These topics will get you to 7+ on the AIME)
Topic Additional Practice and Reference
  • Word Problems.
  • Traditional word problems - Mixture, Work, or Rate, Time, and Distance.
  • Problems in Elementary Mathematics - Lidsky (Algebra.6)
  • Linear and non-linear systems of equations.
  • Equation solving, using symmetry, knowledge of basic identities, factorization, ... .
  • A Problem Book in Algebra - Krechmar (Chapter 1)
  • Logarithms and Exponents.
  • Basic rules and definition, base change, in combination with equation solving and algebraic identities.
  • A Problem Book in Algebra - Krechmar (Chapter 1)
  • Problems in Elementary Mathematics - Lidsky (Algebra.4)
  • Quadratics, Cubics, Polynomials and Vieta's Formulas.
  • Quadratics, Cubics and higher polynomial problems involving roots, coefficients, and Vieta's formulas.
  • Problems in Higher Algebra - Faddeev (Chapter 5.3)
  • Complex Numbers (Basic).
  • Basic complex number questions.
  • Problems in Higher Algebra - Faddeev (Chapter 1.1 and 1.2)
  • Arithmetic and Geometric Sequences and Series (Basic).
  • Basic sequences and series questions generally involving arithmetic and geometric progressions and combinations.
  • Chapter 7 - A Problem Book in Algebra - Krechmar
  • Combinatorial Counting (Basic).
  • Basic combinatorial counting questions and the Binomial Theorem.
  • Counting, 2nd Edition - Meng, Guan (2013)
  • Triangle Geometry (Basic).
  • Application of the most common theorems - Pythagorean, Similar Triangles, Angle Bisector Theorem, Area, Stewart's Theorem, Cevians, Medians, ... .
  • Problems in Elementary Mathematics - Lidsky (Geometry A)
  • Triangle Geometry (Mass Point).
  • Application of all the techniques - Cevians (Medians, Altitudes, Bisectors,) Transversals, Splitting, ratios and in combination with elementary triangle theorems.
  • Number Theory (Elementary).
  • Modular Arithmetic, LCM, GCD, Primes, Base Arithmetic, Integer Algebra, ...
  • Elementary Number Theory: A Problem Oriented Approach - Roberts
  • /
  • Number Theory (Divisors).
  • Application of the most common theorems - Number of Divisors, Sum of Divisors, Product of Divisors, "Revese" problems in each of these areas.
  • Elementary Number Theory: A Problem Oriented Approach - Roberts
  • Probability.
  • Conditional, unconditional, geometric, and elementary combinatorial probabiility questions occur regularly, but not every year.
  • Challenging Mathematics Problems with Elementary Solutions I - Yaglom

To qualify for the USAJMO or USAMO, you will generally need to score 10+ on the AIME.
Olympiad indices the last few years have been quite high, so even if you are scoring 130+ on the AMC, you will still need at least 9 or more to be sure to qualify for the Olympiad.
In order to to do this, you need to study and practice problems and solutions beyond those found in the AoPS books, such as are in the very dense books downloadable below.

AIME 10+ Topics (Add in these to get to 10+)
Topic Additional Advanced Practice and Reference
  • Plane Geometry.
  • Circle theorems, Cicumcircle, Incircle, Ptolemy's Theorem, Power of a Point, various other theorems, and in combination.
  • Problems in Elementary Mathematics - Lidsky (Geometry.A)
  • Problems in Plane Geometry - Sharygin
  • Solid Geometry.
  • There is generally at least one solid geometry problem on the AIME.
  • Problems in Elementary Mathematics - Lidsky (Geometry.B)
  • Problems in Solid Geometry - Sharygin
  • Analytic Geometry.
  • Equations of lines, distance of point to line, equations of circles, ellipses, parabolas, and hyperbolas, area of triangle formulas, ... .
  • Problems in Higher Algebra - Faddeev (Chapter 1.3 and 1.4)
  • Complex Numbers.
  • Advanced complex number questions, roots of unity, the geometry of complex numbers in the Argand plane.
  • Problems in Higher Algebra - Faddeev (Chapter 1.3 and 1.4)
  • Number Theory.
  • Fermat's, Euler's, Wilson's Theorems, linear congruences, Chinese Remainder Theorem, Diophantine Equations, ...
  • Elementary Number Theory: A Problem Oriented Approach - Roberts
  • Polynomials.
  • Polynomial problems involving roots and coefficients, Vieta's formulas, Newton's formulas, symmetric functions.
  • Problems in Higher Algebra - Faddeev (Chapter 5)
  • Combinatorial Counting and Probability.
  • Combinatorial counting and probability questions tend to be of the advanced type, and occur about every other year.
  • Counting, 2nd Edition - Meng, Guan (2013)
  • Principles and Techniques in Combinatorics - Chen, Koh (1992)
  • Triangle Trigonometry.
  • Application of the most common theorems - Law of Cosines, Sines, Tangents, ... combined with elementary theorems, ... .
  • Problems in Elementary Mathematics - Lidsky (Plane Geometry)
  • Trigonometric Equations and Identities.
  • Sum formulas, advanced identities, De Moivre's Theorem, , ... .
  • Problems in Elementary Mathematics - Lidsky (Trigonometry)
  • Sequences and Series.
  • Recursive relation questions.
  • A Problem Book in Algebra - Krechmar (Progressions and Sums)
  • Geometric Modeling.
  • Solving analytic problems with a geometric model.
  • Geometric Modeling
  • Functions.
  • Properties of sin, cos, tan, log, ln, domain and range.
  • AoPS Precalculus
Recommended Problem and Review Books
BookSize
Challenging Problems in Algebra - Posamentier,Salkind 1970 (Dover)  
Challenging Problems in Geometry - Posamentier,Salkind 1970 (Dover)  
Challenging Problems with Elementary Solutions I - Yaglom, Yaglom, 1964 (Dover)  
Problems in Elementary Mathematics - Lidsky 7.3 MB
Problems in Higher Algebra - Faddeev 3.8 MB
A Problem Book in Algebra - Krechmar 9.2 MB
Problems in Plane Geometry - Sharygin (MIR,1982).pdf 14.5 MB
Problems in Solid Geometry - Sharygin (MIR,1986).pdf 3.8 MB
The USSR Olympiad Problem Book - Shklasrsky, Chentzov, and Yaglom 3.2 MB
Elementary Number Theory: A Problem Oriented Approach - Roberts  
Counting, 2nd Edition - Meng, Guan (2013)  
Principles and Techniques in Combinatorics - Chen, Koh (1992)  
More advanced ...  
Mathematical Problems and Puzzles from the Polish Mathematical Olympiads 21.4 MB
250 Problems in Elementary Number Theory - Sierpinski 6.7 MB
   
Please ask one of the coaches for additional study recommendations.

And remember - be respectful to your superiors, if you have any. Good Luck! - DP