Olympiad syllabus.

Core proof methods

• Direct Proof and the Structure of an Argument
• Proof by Contradiction
• Proof by Contrapositive
• The Role of Counterexamples

Writing mathematics

• Writing a Complete Olympiad Solution
• Quantifiers and Logical Flow
• Handling Degenerate and Boundary Cases

Choosing a method

• Contradiction versus Contrapositive
• Proving Equivalence in Two Directions
• Constructive and Non-Constructive Existence

Advanced presentation

• Structuring Multi-Part Solutions
• Partial Results and Their Value

Rigour at olympiad depth

• Minimal Counterexample Arguments
• Detecting Hidden Assumptions
• Writing a Solution a Grader Cannot Fault

Induction

• The Principle of Mathematical Induction
• Strong Induction
• Induction in Divisibility and Inequalities

Recursion

• Recursive Definitions
• Computing Terms of a Recursive Sequence

Structural induction

• Double Induction
• Structural Induction on Configurations
• Induction on the Number of Variables

Well-ordering and descent

• The Well-Ordering Principle
• Infinite Descent
• Descent on a Well-Chosen Measure

Olympiad depth

• Descent in Diophantine Problems
• Termination Arguments in Games and Processes
• Constructing the Right Descent Measure

Core identities

• Difference of Squares and Cubes
• Sum and Difference of Powers
• The Identity x^3 + y^3 + z^3 - 3xyz

Manipulation

• Completing the Square
• Substitution and Normalisation
• Telescoping Expressions

Special factorisations

• The Sophie Germain Identity
• Cyclic and Symmetric Factorisations
• Factoring by Grouping and Adding Terms

Technique

• Normalising a Constraint
• Homogeneous Expressions
• Substitutions that Simplify an Expression

Olympiad depth

• Identity Construction in Hard Problems
• Factorisation as a Descent Tool
• Recognising Hidden Factorisation

First moves

• Understanding the Problem and Restating It
• Small Cases and Pattern Finding
• Working Backwards

Exploration

• Drawing the Right Diagram
• Introducing Notation and Auxiliary Elements

Standard strategies

• The Extremal Principle
• Invariants and Monovariants
• Symmetry Reduction
• Generalisation and Specialisation

Organisation

• Problem Decomposition
• Choosing Between Two Routes

Competition execution

• Time Allocation in a Four and a Half Hour Paper
• Post-Mortem Analysis of Failed Attempts
• Building a Personal Error Log

Polynomial Structure and Division

• Division Algorithm
• Remainder Theorem and Factor Theorem
• Identity Theorem for Polynomials
• Root Structure

Roots, Coefficients and Arithmetic Constraints

• Vieta's Relations
• Rational and Integer Roots
• Polynomials with Integer Values
• Root Location
• Symmetric and Power Sum Methods

Discriminants, Resultants and Common Roots

• Discriminant of a Polynomial
• Resultant and Common Roots
• Reduced Polynomial Problems

Irreducibility and Special Families

• Eisenstein Criterion
• Modular Reduction
• Interpolation and Construction

Elementary symmetric functions

• Elementary Symmetric Functions
• Newton's Identities
• Symmetric Polynomials in Three Variables

Roots of unity

• Roots of Unity and Their Properties
• The Roots of Unity Filter

Structure theorems

• The Fundamental Theorem of Symmetric Polynomials
• Cyclotomic Polynomials
• Chebyshev Polynomials

Reductions

• The uvw Reduction
• The pqr Method

Olympiad depth

• Power Sums and Recurrence Relations
• Constructing Polynomials from Symmetric Data
• Symmetric Techniques in Hard Problems

Standard sequences

• Arithmetic and Geometric Progressions
• Means and Their Properties
• Telescoping Sums

Summation technique

• The Method of Differences
• Sigma Notation and Standard Finite Sums

Advanced summation

• Abel Summation by Parts
• Sums of Powers
• Recursive Sequences and Their Limits

Series

• Convergence of Elementary Series
• Continued-Fraction Convergents as Sequences

Olympiad depth

• Summation Identities in Hard Problems
• Estimating Sums and Asymptotic Bounds
• Generating-Function Proofs of Identities

Foundations

• Order Axioms and Legal Operations
• The Sum of Squares Method
• Equality Cases and Sharpness

Means and convexity

• AM-GM and Weighted AM-GM
• The Power Mean Chain
• Jensen's Inequality and Convexity

Cauchy-Schwarz and its forms

• Cauchy-Schwarz and the Discriminant Proof
• The Engel Form for Fractions
• Holder and Minkowski Inequalities

Sum and product inequalities

• The Rearrangement Inequality
• Chebyshev's Sum Inequality
• Schur's Inequality
• Muirhead and Majorisation

Olympiad depth

• Normalisation of Homogeneous Inequalities
• Ravi Substitution for Triangle Inequalities
• Karamata and Majorisation Arguments
• Sharp Constants and Optimality

Restructuring

• Sum of Squares Certificates
• The Difference Substitution
• Ordering Assumptions and Their Justification

Simple substitutions

• The Substitution a = x/y
• Removing a Product Constraint

Systematic methods

• The uvw Principle
• Smoothing and Mixing Variables
• The Tangent Line Method

Further substitutions

• Ravi Substitution and Generalisations
• Trigonometric Substitutions in Inequalities

Hard applications

• Complete SOS Certificates for Degree-Three Forms
• Boundary Analysis in uvw Problems
• Determining the Sharp Constant
• Discrete and Integer Inequalities

Basic methods

• Substitution and Special Values
• Cauchy-Type Equations
• Functional Equations on Integers and Rationals

Standard families

• Equations Involving f(x + y)
• Equations Involving f(xy)

Structural arguments

• Injectivity and Surjectivity Arguments
• Monotonicity and Boundedness Hypotheses
• Polynomial Solutions of Functional Equations

Two-variable equations

• Equations Involving f(f(x))
• Functional Equations in Two Variables

Olympiad depth

• Hard Functional Equations from Shortlists
• The Regularity Trap and Unstated Assumptions
• Functional Inequalities

Algebra of complex numbers

• Complex Numbers and the Argand Plane
• Modulus, Argument and Polar Form
• De Moivre's Theorem

Roots

• Roots of Unity
• Complex Roots of Polynomial Equations

Geometry with complex numbers

• Lines and Circles in Complex Form
• Concyclic, Collinear and Perpendicular Criteria
• Rotations and Reflections as Multiplication

Classical theorems

• The Complex Form of Classical Theorems
• Sums of Roots of Unity

Olympiad depth

• Hard Geometry by Complex Numbers
• The Complex Proof of Ptolemy's Theorem
• The Triangle Inequality for Moduli

Determinants

• Determinants and Their Properties
• Expansion by Minors
• Determinants in Geometry

Matrices

• Matrix Operations and Inverses
• Solving Systems of Linear Equations

Structure

• Rank and Linear Independence
• Vandermonde and Circulant Determinants

Applications

• Matrices and Linear Recurrences
• Linear Algebra in Combinatorial Problems

Olympiad depth

• Rank Arguments in Combinatorics
• Elementary Use of Eigenvalues
• Determinant Identities and the Trace Trick

Groups in number theory

• Orders of Elements
• Fermat's Little Theorem as a Group Statement
• Cyclic Groups Modulo a Prime

Permutations

• Permutation Groups and Cycle Decomposition
• Parity of Permutations

Group actions

• Group Actions and Orbits
• Burnside's Lemma
• Counting Coloured Configurations

Elementary structure

• Cauchy's Theorem and Elements of Prime Order
• Subgroups and Cosets

Olympiad depth

• Counting Up to Symmetry in Hard Problems
• Group Actions on Sets of Configurations
• Combinatorial Existence by Group Theory

Divisibility

• Divisibility and Its Basic Properties
• Division with Remainder
• Divisibility Rules for Small Numbers
• Divisibility of Sums and Products

Greatest common divisor

• Greatest Common Divisor and Least Common Multiple
• The Euclidean Algorithm
• Bezout's Identity

Extended techniques

• The Extended Euclidean Algorithm
• Linear Combinations and Coprimality
• GCD of Several Numbers and of Polynomial Values

Divisibility structures

• The gcd of x^m - 1 and x^n - 1
• Divisibility of a^n - b^n
• Coprime Integer Pairs and Their Consequences

Olympiad depth

• Divisibility Arguments in Hard Problems
• The GCD of a Set of Binomial Coefficients
• Constructing Divisibility Counterexamples

Hard applications

• Estimating GCDs of Polynomial Values
• Divisibility Chains and Iterated GCDs
• Non-Standard Divisibility Problems

Primes and factorisation

• Prime and Composite Numbers
• The Fundamental Theorem of Arithmetic
• Prime Factorisation Techniques
• Counting Divisors from the Factorisation

Infinitude and distribution

• Euclid's Proof of the Infinitude of Primes
• Primes in Intervals and Bertrand's Postulate
• Primes of Special Forms

Prime structure

• Smooth Numbers and Their Uses
• The Largest Prime Factor of an Integer
• Prime Divisors of Sequences

Quoted results

• Zsigmondy's Theorem as a Quoted Result
• Chebyshev-Type Elementary Bounds
• Using a Deep Theorem Inside a Solution

Olympiad depth

• Primitive Prime Divisors in Hard Problems
• Simultaneous Prime Conditions
• Constructing Integers with Prescribed Prime Structure

Hard applications

• Estimating the Number of Primes in a Range
• Prime Divisors of Factorials and Binomials
• Irreducibility by Prime Arguments

Congruence basics

• Congruences and Their Properties
• Arithmetic with Congruences
• Linear Congruences
• Residue Systems

Core theorems

• Fermat's Little Theorem
• Euler's Theorem
• Wilson's Theorem

Systems and structure

• The Chinese Remainder Theorem
• Non-Coprime Moduli
• The Multiplicative Order of an Element
• Primitive Roots Modulo a Prime

Classical congruences

• Wolstenholme's Theorem
• Quadratic Congruences and Their Solutions
• Congruences Involving Factorials

Olympiad depth

• Order Arguments in Hard Problems
• Constructing Solutions to Congruence Systems
• Congruence Obstructions to Existence

Hard applications

• Exponential Congruences and Index Methods
• Orders Modulo Composite Numbers
• Covering Systems of Congruences

Multiplicative functions

• The Divisor Function tau(n)
• The Sum-of-Divisors Function sigma(n)
• Euler's Totient Function
• Multiplicative and Completely Multiplicative Functions

Elementary results

• The Number of Divisors of a Product
• Sums over Divisors
• Perfect Numbers

Inversion

• The Mobius Function
• Mobius Inversion
• The Dirichlet Convolution

Averages and bounds

• Summatory Estimates for tau and sigma
• Elementary Bounds for phi(n)
• The Average Order of a Divisor Function

Olympiad depth

• Mobius Inversion in Counting Problems
• Multiplicative Functions in Hard Problems
• The Euclid-Euler Characterisation of Perfect Numbers

Hard applications

• Constructing Integers with Prescribed Divisor Data
• Sums Involving phi and mu
• Estimating Divisor Sums Sharply

Linear and quadratic equations

• Linear Diophantine Equations
• The Frobenius Coin Problem
• Pythagorean Triples
• Sums of Two Squares

Method

• Modular Obstructions
• Infinite Descent in Equations
• Bounding and Size Arguments

Classical families

• The Pell Equation
• Vieta Jumping
• The Markov Equation
• Equations Solvable by Factorisation

Deep inputs

• The Sum of Two Squares Theorem
• Gaussian Integers in Diophantine Problems
• Fermat's Last Theorem for Small Exponents

Olympiad depth

• Descent on Quadratic Forms
• Simultaneous Diophantine Conditions
• Exponential Diophantine Equations

Hard applications

• Catalan's Conjecture as a Quoted Result
• Equations with No Small Solutions
• Constructing Infinite Solution Families

Finite continued fractions

• Finite Continued Fractions
• Convergents and Their Recurrence
• Converting a Rational to a Continued Fraction
• The Euclidean Algorithm as a Continued Fraction

Approximation

• Best Rational Approximations
• The Approximation Theorem
• Irrationality Proofs by Approximation

Infinite continued fractions

• Infinite Continued Fractions and Convergence
• Periodic Continued Fractions
• Quadratic Irrationals and Their Periods

Applications

• Solving Pell's Equation by Continued Fractions
• The Hurwitz Approximation Bound
• Approximating Algebraic Numbers

Olympiad depth

• Continued Fractions in Hard Number Theory
• Constructing Good Rational Approximations
• Divisibility Consequences of Approximation

Hard applications

• Simultaneous Approximation Problems
• Continued Fractions and Quadratic Forms
• Estimating the Error of a Convergent

Residues

• Quadratic Residues and Non-Residues
• Euler's Criterion
• The Legendre Symbol
• Counting Solutions of a Quadratic Congruence

Computation

• Computing Legendre Symbols
• The Jacobi Symbol
• Squares Modulo Small Primes

Reciprocity

• The Law of Quadratic Reciprocity
• The Supplementary Laws
• Applying Reciprocity in Problems

Structure

• Primitive Roots and the Structure of Units
• The Quadratic Character of -1, 2 and Small Primes
• Sums of Legendre Symbols

Olympiad depth

• Reciprocity in Hard Number Theory
• Residue Arguments in Diophantine Problems
• Orders and Primitive Roots in Construction Problems

Hard applications

• Quadratic Forms Modulo a Prime
• Proving Non-Residue by Structured Arguments
• Character Sums in Elementary Form

Valuations

• The p-adic Valuation
• Properties of the Valuation
• Valuations of Sums and Products
• Legendre's Formula for v_p(n!)

Divisibility via valuations

• Valuation Arguments in Divisibility
• Comparing Valuations to Prove Bounds
• The Valuation of a Binomial Coefficient

Lifting

• Lifting the Exponent for a^n - b^n
• The Even and Odd Cases of LTE
• Hensel Lifting of a Solution
• Kummer's Theorem on Carries

Applications

• Valuation Bounds in Exponential Equations
• Solving by Comparing Valuations
• Zsigmondy's Theorem in Valuation Form

Olympiad depth

• Exact Valuation Computations in Hard Problems
• Valuation Descent Arguments
• Hard Exponential Diophantine Equations

Hard applications

• Simultaneous Valuation Conditions
• Valuation Tables and Case Analysis
• Defeating a Plausible Solution by Valuation

Sumsets

• Sumsets and Product Sets
• The Size of a Sumset
• Cauchy-Davenport in Small Cases
• Addition Tables Modulo n

Zero sums

• Zero-Sum Problems
• The Erdos-Ginzburg-Ziv Theorem
• Partitioning into Sum-Free Sets

Structural results

• The Cauchy-Davenport Theorem
• The Erdos-Heilbronn Problem
• Sum-Free Sets in the Integers

Quoted results

• Schur's Theorem as a Quoted Result
• Van der Waerden's Theorem as a Quoted Result
• Monochromatic Structures in Colourings

Olympiad depth

• Additive Arguments in Hard Problems
• Zero-Sum Constants and Their Bounds
• Covering Systems and Additive Structure

Hard applications

• Combinatorial Number Theory from Shortlists
• Bounding the Size of a Sumset
• Additive Results Applied to Divisibility

Basic counting

• The Multiplication and Addition Principles
• Permutations and Combinations
• Counting with Repetition
• Arrangements with Restrictions

Structured counting

• Bijections and Double Counting
• Stars and Bars
• Distributions into Boxes

Lattice and arrangement methods

• Lattice Paths
• The Reflection Principle
• Circular and Multiset Arrangements

Special counts

• Derangements
• Counting Onto Functions

Olympiad depth

• Bijection Construction in Hard Problems
• Double Counting Two Ways
• Counting Up to Symmetry

Hard applications

• Extremal Counting Problems
• Counting with Forbidden Patterns

Binomial coefficients

• The Binomial Theorem
• Pascal's Rule
• Properties of Binomial Coefficients
• The Multinomial Theorem

Standard identities

• Vandermonde's Identity
• The Hockey Stick Identity
• Sums of Binomial Coefficients

Proof techniques

• Combinatorial Proof by Double Counting
• Generating-Function Proofs of Identities
• The Roots of Unity Filter

Arithmetic of binomial coefficients

• Lucas's Theorem
• Kummer's Theorem

Olympiad depth

• Hard Binomial Identities from Shortlists
• Alternating Sums and Weighted Identities
• The Chu-Vandermonde Identity

Hard applications

• Binomial Coefficients in Number Theory
• Bounding Binomial Coefficients Sharply

Inclusion-exclusion

• The Inclusion-Exclusion Formula
• Inclusion-Exclusion for Two and Three Sets
• The Complement Method
• Derangements by Inclusion-Exclusion

Counting applications

• Counting Onto Functions
• Counting with Forbidden Positions
• Counting Coprime Integers

Sieve technique

• Legendre's Formula for v_p(n!)
• The Sieve of Eratosthenes as a Counting Device
• Bonferroni-Type Bounds

Weighted forms

• Inclusion-Exclusion with Weights
• Mobius Inversion as a Sieve

Olympiad depth

• Sieve Arguments in Hard Problems
• Partial Sieves with Controlled Error
• Simultaneous Restrictions

Hard applications

• Counting with Dependence Between Properties
• Sharp Bonferroni Constructions

Pigeonhole

• The Pigeonhole Principle
• The Generalised Pigeonhole Principle
• Choosing the Right Boxes
• Pigeonhole Arguments in Geometry

Extremal principle

• The Extremal Principle
• Choosing the Largest or Smallest Object
• Minimal Counterexample Arguments

Structural existence

• Ramsey-Type Existence by Pigeonhole
• The Erdos-Szekeres Monotone Subsequence Theorem
• Lattice Point Arguments

Extremal technique

• Double Counting an Extremal Object
• Degree Arguments in Graphs

Olympiad depth

• Hard Existence Problems
• Forcing Structure by Counting
• Extremal Arguments with Equality Cases

Hard applications

• Dilworth-Type Chain and Antichain Results
• Constructing Counterexamples to Existence Claims

Recurrences

• Setting Up a Recurrence
• Linear Recurrences with Constant Coefficients
• The Characteristic Equation
• The Fibonacci Recurrence

Solving recurrences

• Non-Homogeneous Recurrences
• The Method of Undetermined Coefficients
• Recurrences from Counting Problems

Generating functions

• Ordinary Generating Functions
• Operations on Generating Functions
• The Catalan Generating Function

Applications

• The Transfer Matrix Method
• Counting by Coefficient Extraction

Olympiad depth

• Generating Functions in Hard Counting Problems
• Lagrange Inversion in Elementary Form
• Counting with Obstacles

Hard applications

• Recurrences with Variable Coefficients
• Asymptotics of a Counting Sequence

Graph basics

• Graphs, Vertices and Edges
• Degree and the Handshake Lemma
• Paths, Cycles and Connectivity
• Isomorphism and Subgraphs

Trees and traversals

• Trees and Their Properties
• Spanning Trees
• Eulerian and Hamiltonian Cycles

Structure

• Bipartite Graphs
• Planar Graphs and Euler's Formula
• Graph Colouring

Matchings and trees

• Matchings and Hall's Theorem
• Cayley's Formula for Labelled Trees

Olympiad depth

• Hall's Theorem in Hard Problems
• Menger's Theorem and Connectivity
• Tournament and Orientation Arguments

Hard applications

• Colouring Arguments in Existence Problems
• Building a Graph Model of a Problem

Extremal graph results

• Turan's Theorem
• Triangle-Free Graphs
• Edges Forcing a Complete Subgraph
• Extremal Constructions

Ramsey theory

• Ramsey Numbers
• Ramsey's Theorem for Pairs
• Monochromatic Structures

Further extremal results

• The Kovari-Sos-Turan Theorem
• Zarankiewicz-Type Bounds
• Erdos-Stone as Context

Extremal set theory

• Sperner's Theorem
• The Erdos-Ko-Rado Theorem

Olympiad depth

• Extremal Constructions in Shortlists
• Proving Sharpness of an Extremal Bound
• Ramsey Arguments in Hard Problems

Hard applications

• Extremal Problems on Hypergraphs
• Stability of an Extremal Configuration

Elementary probability

• Sample Spaces and Events
• Conditional Probability
• Independence
• The Law of Total Probability

Expectation

• Random Variables and Expectation
• Linearity of Expectation
• Variance and Covariance

Probabilistic method

• The Probabilistic Method
• Existence by Positive Probability
• Markov's Inequality

Second moment

• Chebyshev's Inequality
• The Second Moment Method

Olympiad depth

• Hard Probabilistic Existence Problems
• The Lovasz Local Lemma in Simple Form
• Expected Value Arguments in Combinatorics

Hard applications

• Alteration Arguments
• Randomised Constructions with Constraints

Linear algebra method

• Linear Independence Arguments
• The Dimension or Rank Bound
• Incidence Matrices
• Applications to Set Systems

Polynomial method

• Polynomials Vanishing on a Set
• The Combinatorial Nullstellensatz
• Applications to Existence Problems

Further algebraic tools

• Chevalley-Warning as a Quoted Result
• The Polynomial Method in Additive Problems
• Eigenvalues of Graphs in Elementary Form

Algebraic certificates

• Fractional Weight Assignments and Averaging Certificates
• Rank Arguments with Symmetry

Olympiad depth

• Hard Algebraic Combinatorics from Shortlists
• Constructing a Matrix Model of a Problem
• Rank Bounds with Equality Analysis

Hard applications

• Polynomial Methods in Number Theory
• Bounding the Size of a Family of Sets

Invariants

• Invariants and Monovariants
• Parity Invariants
• Colouring Arguments for Impossibility
• Termination by a Monovariant

Games

• Winning and Losing Positions
• Backward Induction
• Symmetry and Pairing Strategies

Impartial games

• The Sprague-Grundy Theory
• Nim and Its Variants
• Grundy Values of Simple Games

Strategy arguments

• Strategy Stealing
• Copycat and Potential Function Arguments

Olympiad depth

• Hard Game Problems from Shortlists
• Games on Graphs and Boards
• Determining the Winner by Structural Analysis

Hard applications

• Proving a Draw or a First-Player Win
• Invariants in Iterated Processes

Convexity

• Convex Sets and Convex Hulls
• Vertices and Supporting Lines
• Separating Lines in the Plane
• Convexity in Combinatorial Arguments

Lattice geometry

• Lattice Points and Lattice Polygons
• Pick's Theorem
• Counting Lattice Points in a Region

Classical results

• Helly's Theorem
• Radon's Theorem
• Caratheodory's Theorem

Geometric counting

• Minkowski's Lattice Point Theorem
• Covering and Packing in the Plane

Olympiad depth

• Hard Combinatorial Geometry from Shortlists
• Convex Hull Arguments in Extremal Problems
• Lattice Point Arguments in Number Theory

Hard applications

• Bounding the Number of Unit Distances
• Incidence Problems in the Plane

Centres of a triangle

• The Circumcentre and the Circumcircle
• The Incentre and the Incircle
• The Orthocentre
• The Centroid

Elementary triangle results

• The Angle Bisector Theorem
• The Sine and Cosine Rules in Geometry
• The Area Formulae
• Congruence and Similarity Criteria

Classical lines and circles

• The Euler Line
• The Nine-Point Circle
• The Feuerbach Theorem
• The Excentres and Excircles

Metric relations

• Ceva's Theorem
• Menelaus's Theorem
• Stewart's Theorem
• Apollonius's Theorem

Advanced triangle geometry

• Symmedians and the Lemoine Point
• The Gergonne and Nagel Points
• The Trigonometric Form of Ceva and Menelaus
• Coordinates of the Classical Centres

Olympiad configurations

• Hard Triangle Problems from Shortlists
• The Orthic Triangle and Its Properties
• The Intouch Triangle

Angles and circles

• The Inscribed Angle Theorem
• Angles in the Same Segment
• The Tangent-Chord Angle
• Cyclic Quadrilateral Criteria

Power and tangency

• The Power of a Point
• Intersecting Chords and Secants
• The Tangent-Secant Theorem
• Lengths of Tangents

Radical structure

• The Radical Axis
• The Radical Centre
• Coaxal Systems of Circles
• Orthogonal Circles

Classical cyclic results

• Ptolemy's Theorem
• The Ptolemy Inequality
• The Simson Line
• Miquel's Theorem

Advanced circle geometry

• The Butterfly Theorem
• Casey's Theorem
• Spiral Similarity in Circle Problems
• The Complete Quadrilateral and Its Circles

Olympiad configurations

• Hard Circle Problems from Shortlists
• The Mixtilinear Incircle
• Tangent and Secant Configurations

Isometries

• Translation and Rotation
• Reflection and Glide Reflection
• Composition of Isometries
• Symmetry of a Figure

Similarity

• Homothety and Its Centre
• Direct Similarity
• Indirect Similarity
• The Composition of Two Homotheties

Spiral similarity

• Spiral Similarity and Its Centre
• Locating the Centre of a Spiral Similarity
• Spiral Similarity in Two-Circle Problems
• Similar Triangles on the Sides of a Triangle

Transformation technique

• Solving a Problem by a Transformation
• The Method of the Moving Point
• Rotation by 60 and 90 Degrees
• Identifying the Invariant Figure

Advanced transformations

• The Homothety of the Incircle and the Excircle
• The Nine-Point Circle by Homothety
• Transformations in Hard Shortlist Problems
• Composition and Group Structure of Transformations

Olympiad configurations

• Miquel Points by Spiral Similarity
• Napoleon-Type Configurations
• The Simson Line by Rotation

Inversion

• Inversion in a Circle and Its Definition
• The Inversion Distance Formula
• Inversion of a Line
• Inversion of a Circle

First applications

• Inversion and Tangency
• Inversion Preserving Angles
• Inversion of a Circle Through the Centre
• Choosing the Inversion Circle

Inversion technique

• Circles Orthogonal to the Inversion Circle
• The Inversion of a Triangle
• Inversion and the Radical Axis
• The Standard Inversion Configuration Catalogue

Advanced circle techniques

• Spiral Similarity with Inversion
• The Archimedes Twin Circles
• Pappus Chains and Arbelos Configurations
• The Shoemaker's Knife

Advanced inversion

• Hard Inversion Problems from Shortlists
• Inversion Combined with Homothety
• The Feuerbach Theorem by Inversion
• Inversion in Three Dimensions

Olympiad configurations

• Casey's Theorem by Inversion
• Tangent Circle Chains
• Constructing the Required Inversion

Poles and polars

• The Polar of a Point with Respect to a Circle
• La Hire's Theorem
• The Pole of a Line
• Constructing a Polar

Classical projective facts

• Harmonic Conjugates
• The Complete Quadrilateral
• Cross Ratio and Its Invariance
• The Projective Line

Projective theorems

• Desargues's Theorem
• Pascal's Theorem
• Brianchon's Theorem
• The Projective Viewpoint on Classical Theorems

Projective technique

• Projective Transformations
• Sending a Conic to a Circle
• Proving a Collinearity Projectively
• Proving a Concurrency Projectively

Advanced projective geometry

• Hard Projective Problems from Shortlists
• Poles and Polars for a General Conic
• The Projective Theory of the Complete Quadrilateral
• Involutions on a Line

Olympiad configurations

• Pascal Applied to Degenerate Hexagons
• The Polar of a Point on a Conic
• Brokard's Theorem

Coordinate methods

• Cartesian Coordinates in Geometry
• The Distance and Area Formulae in Use
• Lines and Circles in Coordinates
• Solving a Configuration by Coordinates

Vector methods

• Vectors in the Plane
• The Dot Product and Perpendicularity
• The Cross Product and Area
• Vector Proofs of Classical Results

Barycentric and trilinear coordinates

• Barycentric Coordinates
• Trilinear Coordinates
• Coordinates of the Classical Centres
• The Equation of a Line in Barycentrics

Complex methods

• The Complex Plane and the Unit Circle
• Complex Criteria for Collinearity and Concyclicity
• Rotations as Complex Multiplication
• The Complex Form of Classical Theorems

Advanced analytic geometry

• Hard Geometry by Complex Numbers
• The Complex Proof of Ptolemy's Theorem
• Computing with Named Triangle Centres
• Choosing the Right Coordinate System

Olympiad configurations

• Proving a Concyclicity by Computation
• Algebraic Proofs of Collinearity
• Hybrid Synthetic and Analytic Solutions

Trigonometric fundamentals

• Radian Measure and the Unit Circle
• The Sine and Cosine Rules
• Trigonometric Identities for Geometry
• The Area Formulae in Trigonometric Form

Triangle trigonometry

• The Half-Angle Formulae
• The Inradius and Circumradius Formulae
• Solving a Triangle
• Heights and Distances

Trigonometric technique in geometry

• The Tangent Half-Angle Substitution
• The Trigonometric Form of Ceva and Menelaus
• The Auxiliary Angle Method
• Trigonometric Substitutions in Optimisation

Trigonometric sums

• Sums of Sines and Cosines in Arithmetic Progression
• The Roots of Unity Filter in Trigonometric Form
• The Lagrange Trigonometric Identity
• Chebyshev Polynomials in Trigonometric Form

Advanced trigonometric methods

• Hard Trigonometric Problems from Shortlists
• Trigonometric Inequalities
• Bounds for Trigonometric Sums
• Trigonometric Parametrisation of a Configuration

Olympiad configurations

• Angle Chasing with Trigonometric Identities
• Proving an Identity in a Triangle
• Trigonometric Solutions of Optimisation Problems

Solids and their measures

• The Tetrahedron and Its Volumes
• The Sphere, Cone and Cylinder
• Surface Area and Volume Formulae
• Sections of a Solid

Three-dimensional configurations

• Angles Between Lines and Planes
• The Dihedral Angle
• The Distance Between Skew Lines
• Perpendicularity in Space

Structure of polyhedra

• Euler's Polyhedron Formula
• The Platonic Solids
• Polyhedra and Planar Graphs
• Spheres Inscribed in and Circumscribed about a Polyhedron

Technique

• Volumes by Decomposition
• Cavalieri's Principle
• Vectors and Coordinates in Space
• The Method of Projection onto a Plane

Advanced solid geometry

• Hard Three-Dimensional Problems from Shortlists
• The Geometry of the Regular Tetrahedron
• Spherical Sections and Great Circles
• Optimisation Problems in Space

Olympiad configurations

• The Sphere Through Four Points
• Three-Dimensional Lattice Problems
• Reducing a Space Problem to a Plane

Basic geometric inequalities

• The Triangle Inequality in Geometry
• Area and Perimeter Relations
• The Isoperimetric Inequality
• Inequalities Involving the Inradius and Circumradius

Technique

• Geometric Optimisation by Symmetry
• Reflection and Shortest Paths
• The Method of the Extremal Configuration
• Reduction to an Algebraic Inequality

Classical triangle inequalities

• Weitzenbock's Inequality
• Euler's Inequality R >= 2r
• The Erdos-Mordell Theorem
• The Neuberg-Pedoe Inequality

Extremal geometry

• Extremal Problems for Polygons
• Covering and Packing Problems
• The Largest or Smallest Configuration
• Equality Cases in Geometric Inequalities

Advanced geometric inequalities

• Hard Geometric Inequalities from Shortlists
• The Finsler-Hadwiger Inequality
• The Ono Inequality
• Inequalities for a Tetrahedron

Olympiad configurations

• Sharpening an Inequality by a Better Lemma
• Proving Sharpness Geometrically
• Combining Geometric and Algebraic Inequalities