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
- Argument by Contradiction
- 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
