Complete Olympiad & Competition Mathematics Mastery Program

📚 Topics Covered

• Competition mathematics vs school mathematics
• Major competitions overview: AMC, AIME, USAMO, IMO
• Competition formats and scoring systems
• Problem-solving strategies: working backwards
• Pattern recognition and generalization
• Extreme cases and boundary testing
• Proof by contradiction
• Proof by induction basics
• Direct proof techniques
• Constructive vs non-constructive proofs

• Mathematical notation for competitions
• Time management in competitions

📚 Topics Covered

• Advanced divisibility rules and proofs
• GCD and LCM properties and algorithms
• Euclidean algorithm and extended version
• Bezout's identity and applications
• Prime factorization uniqueness theorem
• Distribution of primes
• Sieve of Eratosthenes variations
• Prime number theorem introduction
• Fermat's Little Theorem
• Wilson's Theorem

• Prime testing algorithms
• Applications in cryptography basics

📚 Topics Covered

• Modular arithmetic operations and properties
• Congruence classes and equivalence
• Linear congruences and solutions
• Chinese Remainder Theorem
• Modular multiplicative inverse
• Euler's totient function φ(n)
• Euler's theorem and generalizations
• Quadratic residues introduction
• Legendre symbol
• Power patterns in modular arithmetic

• Solving Diophantine equations
• Applications to competition problems

📚 Topics Covered

• Perfect numbers and Mersenne primes
• Arithmetic functions: σ(n), τ(n), μ(n)
• Multiplicative functions
• Mobius inversion formula
• Continued fractions basics
• Pell's equation
• Quadratic reciprocity introduction
• Sum of squares theorems
• Lifting the Exponent Lemma (LTE)
• Zsigmondy's theorem

• Carmichael's lambda function
• Advanced competition techniques

📚 Topics Covered

• Advanced counting principles
• Multiplication and addition principles
• Permutations with restrictions
• Circular permutations and necklaces
• Combinations with repetition
• Stars and bars method
• Inclusion-Exclusion Principle
• Derangements and subfactorial
• Stirling numbers of both kinds
• Partition function introduction

• Generating functions basics
• Recurrence relations

📚 Topics Covered

• Graph definitions and terminology
• Types of graphs: complete, bipartite, planar
• Degree sequences and handshaking lemma
• Trees and forest properties
• Eulerian paths and circuits
• Hamiltonian paths and cycles
• Graph coloring and chromatic number
• Planar graphs and Euler's formula
• Bipartite matching basics
• Hall's marriage theorem

• Tournament graphs
• Competition graph problems

📚 Topics Covered

• Number theory comprehensive review
• Modular arithmetic mastery check
• Combinatorics problem-solving strategies
• Graph theory applications
• Mixed problem sets
• Speed solving techniques
• Competition simulation

📚 Topics Covered

• Algebraic manipulation mastery
• Telescoping sums and products
• Partial fractions in competitions
• Symmetric expressions and polynomials
• Vieta's formulas and applications
• Newton's identities
• Power sum symmetric functions
• Discriminant and its properties
• Algebraic substitutions techniques
• Rationalizing complex expressions

• Nested radicals simplification
• Competition algebraic tricks

📚 Topics Covered

• Polynomial division and remainder theorem
• Factor theorem and applications
• Rational root theorem extensions
• Descartes' rule of signs
• Complex roots and conjugate pairs
• Fundamental theorem of algebra
• Polynomial interpolation (Lagrange)
• Chebyshev polynomials introduction
• Cyclotomic polynomials
• Irreducibility testing

• Polynomial inequalities
• Roots and coefficients relationships

📚 Topics Covered

• Arithmetic and geometric progressions advanced
• Arithmetico-geometric progressions
• Harmonic progressions and means
• Recursive sequences and linear recurrences
• Characteristic equations method
• Generating functions for sequences
• Fibonacci and Lucas sequences properties
• Catalan numbers and applications
• Convergence tests for series
• Telescoping series mastery

• Power series basics
• Competition sequence problems

📚 Topics Covered

• Systems of linear equations (advanced)
• Nonlinear systems solving techniques
• Symmetric systems of equations
• Cyclic systems
• Homogeneous equations
• Parametric solutions
• Diophantine equations (advanced)
• Quadratic Diophantine equations
• Simon's Favorite Factoring Trick
• Equations with absolute values

• Floor and ceiling function equations
• Competition equation strategies

📚 Topics Covered

• Complex number operations review
• Geometric interpretation of complex operations
• De Moivre's theorem and applications
• Roots of unity and cyclotomic polynomials
• Complex number proofs in geometry
• Trigonometric identities via complex numbers
• Gaussian integers
• Complex conjugate root theorem
• Argument and modulus inequalities
• Competition problems using complex numbers

• Euler's formula applications
• Advanced complex number techniques

📚 Topics Covered

• Introduction to functional equations
• Cauchy's functional equations
• Substitution methods in functional equations
• Finding solutions: injectivity, surjectivity
• Additive and multiplicative functions
• Jensen's functional equation
• D'Alembert's functional equation
• Polynomial functional equations
• Functional equations with multiple variables
• Cyclic functional equations

• IMO functional equation techniques
• Common competition patterns

📚 Topics Covered

• Strong induction principles
• Structural induction
• Double induction
• Infinite descent method
• Well-ordering principle applications
• Induction with inequalities
• Induction in number theory
• Induction in combinatorics
• Induction for sequences
• Forward-backward induction

• Transfinite induction introduction
• Competition induction strategies

📚 Topics Covered

• Pigeonhole principle variations
• Generalized pigeonhole principle
• Infinite pigeonhole principle
• Probabilistic pigeonhole principle
• Dirichlet's theorem applications
• Extremal principle in problem solving
• Optimization in discrete mathematics
• Erdős-Ko-Rado theorem introduction
• Ramsey theory basics
• Van der Waerden's theorem introduction

• Extremal graph theory
• Competition applications

📚 Topics Covered

• Invariant principle in problem solving
• Finding invariants in processes
• Monovariant strategies
• Parity as an invariant
• Coloring arguments
• Sum and product invariants
• Geometric invariants
• Game theory and invariants
• Invariants in combinatorial games
• Semi-invariants

• Competition invariant problems
• Creating invariant arguments

📚 Topics Covered

• Algebra and polynomials review
• Sequences and series mastery
• Functional equations practice
• Complex numbers applications
• Induction and extremal principles
• Mixed advanced problems
• AIME simulation

📚 Topics Covered

• Cevians: medians, altitudes, angle bisectors
• Special points: centroid, orthocenter, incenter, circumcenter
• Euler line and nine-point circle
• Nagel point and Gergonne point
• Stewart's theorem and applications
• Menelaus' theorem
• Ceva's theorem and trigonometric form
• Mass point geometry
• Barycentric coordinates introduction
• Triangle inequality variations

• Routh's theorem
• Competition triangle problems

📚 Topics Covered

• Power of a point theorem
• Radical axis and radical center
• Coaxial circles
• Inversion in circles
• Ptolemy's theorem and inequality
• Cyclic quadrilaterals properties
• Simson line and pedal triangles
• Miquel's theorem
• Pascal's theorem for circles
• Brianchon's theorem

• Nine-point circle properties
• Competition circle techniques

📚 Topics Covered

• Advanced similarity principles
• Spiral similarities
• Homothety and homothetic centers
• Similar triangles in competitions
• Angle bisector theorem extensions
• Apollonius' theorem
• Harmonic division and conjugates
• Cross-ratio and applications
• Desargues' theorem
• Monge's theorem

• Homothety in circle problems
• Competition applications

📚 Topics Covered

• Reflection properties and compositions
• Rotation compositions and fixed points
• Translation vectors and compositions
• Glide reflections
• Isometries classification
• Dilation and scaling transformations
• Affine transformations
• Transformation approach to problems
• Symmetry in problem solving
• Spiral similarities revisited

• Complex numbers for transformations
• Competition transformation techniques

📚 Topics Covered

• Classical constructions review
• Constructible numbers theory
• Impossible constructions and proofs
• Mohr-Mascheroni constructions
• Poncelet-Steiner constructions
• Construction strategies in competitions
• Locus problems
• Construction with restrictions
• Geometric optimization constructions
• Construction existence proofs

• Competition construction problems
• Software-aided exploration

📚 Topics Covered

• Trigonometric identities mastery
• Sum-to-product and product-to-sum formulas
• Multiple angle formulas
• Half-angle formulas applications
• Conditional identities
• Trigonometric equations (advanced)
• Trigonometric inequalities
• Jensen's inequality for trigonometry
• Chebyshev polynomials and trigonometry
• Trigonometric substitutions

• Competition trigonometry tricks
• Proofs using trigonometry

📚 Topics Covered

• Coordinate geometry strategies
• Distance and section formulas
• Area calculations in coordinates
• Line equations and properties
• Angle between lines
• Conic sections: parabola, ellipse, hyperbola
• Parametric representations
• Polar coordinates applications
• Coordinate transformations
• Analytic proofs of geometric theorems

• Optimization using coordinates
• Competition coordinate techniques

📚 Topics Covered

• Vector operations and properties
• Dot product and cross product
• Vector proofs of geometric theorems
• Position vectors and applications
• Vector equations of lines and planes
• Distance using vectors
• Angle calculations with vectors
• Vector approach to transformations
• Centroid and circumcenter via vectors
• 3D geometry basics

• Competition vector techniques
• Vectors vs coordinates comparison

📚 Topics Covered

• Area calculation techniques
• Shoelace formula and applications
• Pick's theorem
• Heron's formula and extensions
• Bretschneider's formula
• Brahmagupta's formula
• Volume calculation methods
• Cavalieri's principle
• Volume by cross-sections
• Surface area calculations

• Pappus's centroid theorems
• Competition area/volume tricks

📚 Topics Covered

• Euclidean geometry mastery review
• Circle theorems compilation
• Transformation techniques
• Trigonometry applications
• Coordinate methods review
• Vector approaches
• Mixed geometry problems
• USAMO geometry preparation

📚 Topics Covered

• AM-GM inequality and applications
• Cauchy-Schwarz inequality proofs and uses
• Holder's inequality
• Minkowski's inequality
• Power mean inequalities
• Weighted AM-GM inequality
• Jensen's inequality for convex functions
• Karamata's inequality
• Rearrangement inequality
• Chebyshev's inequality

• Bernoulli's inequality
• Competition inequality techniques

📚 Topics Covered

• Smoothing and majorization
• Mixing variables technique
• Lagrange multipliers introduction
• Homogenization techniques
• Substitutions in inequalities
• Cyclic and symmetric inequalities
• Buffalo Way method
• SOS (Sum of Squares) method
• Schur's inequality and generalizations
• Muirhead's inequality

• Newton's inequalities
• IMO inequality strategies

📚 Topics Covered

• Discrete optimization techniques
• Continuous optimization basics
• Calculus in olympiad problems
• Derivatives for optimization
• Constrained optimization
• Geometric optimization problems
• Isoperimetric problems
• Algebraic optimization
• Combinatorial optimization
• Game theory optimization

• Dynamic programming introduction
• Competition optimization strategies

📚 Topics Covered

• Bijective proofs and counting
• Double counting techniques
• Combinatorial identities (advanced)
• Vandermonde's identity applications
• Catalan number applications
• Bell numbers and Stirling numbers
• Integer partitions
• Young tableaux introduction
• Polya enumeration theorem
• Combinatorial game theory

• Probabilistic methods
• Competition combinatorics mastery

📚 Topics Covered

• Quadratic reciprocity (complete)
• Primitive roots and indices
• Quadratic forms basics
• Continued fractions (advanced)
• Transcendental numbers introduction
• p-adic numbers introduction
• Analytic number theory glimpse
• Additive number theory basics
• Beatty sequences
• Thue's lemma

• Advanced Diophantine equations
• IMO number theory trends

📚 Topics Covered

• IMO problem structure and trends
• Problem 1 and 4 strategies (easier problems)
• Problem 2 and 5 strategies (medium problems)
• Problem 3 and 6 strategies (hardest problems)
• Time management for 4.5-hour exams
• Partial credit maximization
• Problem selection strategies
• Writing rigorous proofs
• Common IMO problem patterns
• Historical IMO problems analysis

• National olympiad comparison
• Training camp preparation

📚 Topics Covered

• Putnam exam format and scoring
• Putnam problem characteristics
• Abstract algebra in Putnam
• Real analysis topics
• Linear algebra applications
• Probability in Putnam problems
• Putnam trick techniques
• Time management for Putnam
• A vs B session strategies
• Historical Putnam analysis

• University-level competition transition
• Research mathematics introduction

📚 Topics Covered

• APMO problem characteristics
• Regional competition differences
• APMO vs IMO difficulty comparison
• Cultural problem-solving approaches
• APMO-specific techniques
• Time zone competition strategies
• Online competition protocols
• APMO historical trends
• Country-specific olympiad styles
• International collaboration benefits

• APMO as IMO preparation
• Regional olympiad circuit

📚 Topics Covered

• Mental preparation for competitions
• Stress management techniques
• Visualization and positive thinking
• Competition day routines
• Dealing with difficult problems
• Recovery from mistakes
• Time pressure management
• Building mathematical confidence
• Team competition strategies
• Learning from competition failures

• Post-competition analysis
• Long-term training mindset

📚 Topics Covered

• From olympiads to research mathematics
• Open problems in mathematics
• Reading mathematical papers
• Mathematical writing skills
• LaTeX for mathematics
• Undergraduate research opportunities
• REU programs and applications
• Mathematical journals for students
• Online mathematics communities
• Collaboration in mathematics

• Career paths in mathematics
• Graduate school preparation

📚 Topics Covered

• USAMO complete simulation
• USAJMO preparation strategies
• Canadian Mathematical Olympiad
• British Mathematical Olympiad
• Australian Mathematics Olympiad
• Indian National MO preparation
• China National Team Selection
• Russia Mathematical Olympiad
• Time management across formats
• Scoring system understanding

• Selection criteria for teams
• National camp preparation

📚 Topics Covered

• IMO Day 1 simulation (4.5 hours)
• IMO Day 2 simulation (4.5 hours)
• Complete solution writing
• Proof verification techniques
• Collaboration after competition
• Score prediction and analysis
• International competition etiquette
• Cultural exchange benefits
• Language considerations
• Travel and preparation logistics

• Team dynamics
• Medal boundaries analysis

📚 Topics Covered

• Creating original problems
• Problem difficulty assessment
• Multi-topic problem design
• Solution elegance principles
• Alternative solution finding
• Problem generalization techniques
• Special case analysis
• Problem posing for competitions
• Reviewing peer solutions
• Mathematical exposition

• Building problem databases
• Contributing to mathematics community

📚 Topics Covered

• Probabilistic method in combinatorics
• Algebraic topology glimpses
• Category theory basics
• Model theory introduction
• Ergodic theory applications
• Representation theory basics
• Algebraic geometry introduction
• Differential geometry concepts
• Measure theory basics
• Fourier analysis glimpses

• Mathematical physics connections
• Frontier mathematics overview

📚 Topics Covered

• Number theory complete review
• Combinatorics mastery assessment
• Algebra and polynomials review
• Geometry techniques compilation
• Inequality mastery check
• Functional equations review
• Problem-solving strategies synthesis
• Proof writing excellence
• Speed and accuracy balance
• Knowledge gap identification

• Personal strength analysis
• Improvement planning

📚 Topics Covered

• AMC 12 simulation
• AIME simulation
• USAMO simulation Day 1
• USAMO simulation Day 2
• IMO simulation Day 1
• IMO simulation Day 2
• Score analysis and prediction
• Performance evaluation
• Peer competition and ranking
• Solution presentation

• Awards and recognition
• Competition portfolio completion

📚 Topics Covered

• University mathematics programs
• Scholarship opportunities
• Research program applications
• Summer mathematics camps
• International exchange programs
• Online competition platforms
• Mathematical communities joining
• Mentorship opportunities
• Teaching and tutoring paths
• Competition coaching possibilities

• Long-term mathematics goals
• Life-long learning strategies