 Putnam and Beyond Advanced book on Mathematics Olympiad by R˘azvan Gelca and Titu Andreescu

## Contents of Putnam and Beyond Mathematics Olympiad

• A Study Guide
• Methods of Proof
• Mathematical Induction
• The Pigeonhole Principle
• Ordered Sets and Extremal Elements
• Invariants and Semi-Invariants
• Algebra
• Identities and Inequalities
• Algebraic Identities
• The Cauchy–Schwarz Inequality
• The Triangle Inequality
• The Arithmetic Mean–Geometric Mean Inequality
• Sturm’s Principle
• Other Inequalities
• Polynomials
• A Warmup
• Viète’s Relations
• The Derivative of a Polynomial
• The Location of the Zeros of a Polynomial
• Irreducible Polynomials
• Chebyshev Polynomials
• Linear Algebra
• Operations with Matrices
• Determinants
• The Inverse of a Matrix
• Systems of Linear Equations
• Vector Spaces, Linear Combinations of Vectors, Bases
• Linear Transformations, Eigenvalues, Eigenvectors
• The Cayley–Hamilton and Perron–Frobenius Theorems
• Abstract Algebra
• Binary Operations
• Groups
• Rings
• Real Analysis
• Sequences and Series
• Search for a Pattern
• Linear Recursive Sequences
• Limits of Sequences
• More About Limits of Sequences
• Series
• Telescopic Series and Products
• Continuity, Derivatives, and Integrals
• Limits of Functions
• Continuous Functions
• The Intermediate Value Property
• Derivatives and Their Applications
• The Mean Value Theorem
• Convex Functions
• Indefinite Integrals
• Definite Integrals
• Riemann Sums
• Inequalities for Integrals
• Taylor and Fourier Series
• Multivariable Differential and Integral Calculus
• Partial Derivatives and Their Applications
• Multivariable Integrals
• The Many Versions of Stokes’ Theorem
• Equations with Functions as Unknowns
• Functional Equations
• Ordinary Differential Equations of the First Order
• Ordinary Differential Equations of Higher Order
• Problems Solved with Techniques of Differential Equations
• Geometry and Trigonometry
• Geometry
• Vectors
• The Coordinate Geometry of Lines and Circles
• Conics and Other Curves in the Plane
• Coordinate Geometry in Three and More Dimensions
• Integrals in Geometry
• Other Geometry Problems
• Trigonometry
• Trigonometric Identities
• Euler’s Formula
• Trigonometric Substitutions
• Telescopic Sums and Products in Trigonometry
• Number Theory
• Integer-Valued Sequences and Functions
• Some General Problems
• Fermat’s Infinite Descent Principle
• The Greatest Integer Function
• Arithmetic
• Factorization and Divisibility
• Prime Numbers
• Modular Arithmetic
• Fermat’s Little Theorem
• Wilson’s Theorem
• Euler’s Totient Function
• The Chinese Remainder Theorem
• Diophantine Equations
• Linear Diophantine Equations
• The Equation of Pythagoras
• Pell’s Equation
• Other Diophantine Equations
• Combinatorics and Probability
• Combinatorial Arguments in Set Theory and Geometry
• Set Theory and Combinatorics of Sets
• Permutations
• Combinatorial Geometry
• Euler’s Formula for Planar Graphs
• Ramsey Theory
• Binomial Coefficients and Counting Methods
• Combinatorial Identities
• Generating Functions
• Counting Strategies
• The Inclusion–Exclusion Principle
• Probability
• Equally Likely Cases
• Establishing Relations Among Probabilities
• Geometric Probabilities
• Solutions
• Methods of Proof
• Algebra
• Real Analysis
• Geometry and Trigonometry
• Number Theory
• Combinatorics and Probability

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