Olympiad Mathematics

Book 1. Introduction to Olympiad Number Theory

• 1. Divisibility and Prime Factorisation
• 2. GCD, LCM and Euclidean Algorithm
• 3. Modular Arithmetic I: Residues and Contradictions
• 4. Modular Arithmetic II: Linear Congruences and Systems
• 5. Diophantine Equations I: Factorisation and Bounds
• 6. Infinite Descent I
• 7. Fermat, Euler and Power Cycles
• 8. Chinese Remainder Theorem
• 9. Divisor Counting
• 10. Digits, Bases and Periodicity
• 11. Mixed Problems I
• 12. Mock Olympiads I

Book 1. Foundations of Olympiad Algebra

• 1. Algebraic Expressions and Identities
• 2. Factorisation Methods
• 3. Equations and Systems
• 4. Polynomials I
• 5. Sequences and Recurrences
• 6. Algebraic Transformations in Problems
• 7. Introductory Inequalities
• 8. Introductory Functional Equations
• 9. Algebraic Number Problems
• 10. Mixed Algebra Problems I
• 11. Strategy Notes
• 12. Mock Olympiads I

Book 1. Foundations of Olympiad Geometry

• 1. Angles, Lines and Parallel Lines
• 2. Triangles I: Congruence
• 3. Triangles II: Similarity
• 4. Quadrilaterals
• 5. Circles I: Basic Circle Geometry
• 6. Areas I
• 7. Basic Constructions and Auxiliary Lines
• 8. Mixed Problems I

Book 1. Foundations of Olympiad Combinatorics

• 1. Counting Principles
• 2. Permutations and Arrangements
• 3. Combinations
• 4. Counting in Two Ways
• 5. Pigeonhole Principle I
• 6. Invariants I
• 7. Coloring and Board Problems
• 8. Games and Strategies I
• 9. Graphs I
• 10. Recursion and Sequences
• 11. Mixed Problems I
• 12. Mock Olympiads I

Book 2. Olympiad Number Theory Methods

• 1. Advanced GCD Problems
• 2. Quadratic Residues and Modular Obstructions
• 3. Multiplicative Order
• 4. Wilson, Fermat and Euler in Problems
• 5. p-adic Valuations
• 6. LTE: Lifting the Exponent
• 7. Diophantine Equations I: Factorisation and Bounds
• 8. Diophantine Equations II: Descent and Vieta Jumping
• 9. Chinese Remainder Theorem and Construction
• 10. Arithmetic Functions
• 11. Digits, Bases and Decimal Periods
• 12. Polynomials, Sequences and Number Theory

Book 2. Olympiad Inequalities

• 1. Basic Inequality Principles
• 2. AM-GM
• 3. Cauchy-Schwarz
• 4. Rearrangement and Chebyshev
• 5. Jensen's Inequality Intro
• 6. UVW Method and Symmetric Inequalities
• 7. Homogeneous Inequalities
• 8. Substitution Methods
• 9. Inequalities with Constraints
• 10. Hard Inequality Problems
• 11. Mixed Inequality Sets
• 12. Mock Olympiads: Inequalities

Book 3. Functional Equations

• 1. What Is a Functional Equation?
• 2. First Substitutions
• 3. Linear Functional Equations
• 4. Injectivity and Surjectivity
• 5. Cauchy-Type Equations
• 6. Functional Equations on Integers
• 7. Polynomial Functional Equations
• 8. Iteration
• 9. Inequality Conditions in Functional Equations
• 10. Advanced Functional Equations
• 11. Mixed Functional Equations

Book 2. Olympiad Geometry Methods

• 1. Advanced Angle Chasing
• 2. Power of a Point
• 3. Radical Axis
• 4. Homothety and Spiral Similarity
• 5. Inversion I: First Contact
• 6. Ceva and Menelaus
• 7. Area Method II
• 8. Complete Quadrilaterals and Miquel Points
• 9. Geometry with Coordinates and Vectors
• 10. Mixed Problems II

Book 3. Advanced Olympiad Geometry

• 1. Inversion II
• 2. Projective Geometry I
• 3. Poles and Polars
• 4. Simson Line and Pedal Geometry
• 5. Brocard, Napoleon, and Special Points
• 6. Trigonometric Geometry

Book 2. Olympiad Combinatorics Methods

• 1. Advanced Double Counting
• 2. Inclusion-Exclusion
• 3. Bijections and Encoding Objects
• 4. Extremal Principle
• 5. Invariants II and Monovariants
• 6. Ramsey-Type Ideas
• 7. Graphs II
• 8. Matching and Hall's Theorem Intro
• 9. Generating Functions I