- Chapter
- Chapter
- Chapter
- What This Book Is About and How to Read It
- 1 .1 "Exercises" vs. "Problems"
- 1.2 The Three Levels of Problem Solving
- 1.3 A Problem Sampler
- 1.4 How to Read This Book
- Strategies for Investigating Problems
- 2.1 Psychological Strategies
- Mental Toughness: Learn from Polya's Mouse
- Creativity
- 2.2 Strategies for Getting Started
- The First Step: Orientation
- I'm Oriented. Now What?
- 2.3 Methods of Argument
- Common Abbreviations and Stylistic Conventions
- Deduction and Symbolic Logic
- Argument by Contradiction
- Mathematical Induction
- 2.4 Other Important Strategies
- Draw a Picture!
- Pictures Don't Help? Recast the Problem in Other Ways!
- Change Your Point of View
- Tactics for Solving Problems
- 3.1 Symmetry
- Geometric Symmetry
- Algebraic Symmetry
- 3.2 The Extreme Principle
- 3.3 The Pigeonhole Principle
- Basic Pigeonhole
- Intermediate Pigeonhole
- Advanced Pigeonhole
- Chapter xvi CONTENTS
- Chapter
- 3.4 Invariants
- Parity
- Modular Arithmetic and Coloring
- Monovariants
- Three Important Crossover Tactics
- 4.1 Graph Theory
- Connectivity and Cycles
- Eulerian and Hamiltonian Paths
- The Two Men of Tibet
- 4.2 Complex Numbers
- Basic Operations
- Roots of Unity
- Some Applications
- 4.3 Generating Functions
- Introductory Examples
- Recurrence Relations
- Partitions
- Algebra
- Basic Operations
- 5.1 Sets, Numbers, and Functions
- Sets
- Functions
- 5.2 Algebraic Manipulation Revisited
- The Factor Tactic
- Manipulating Squares
- Substitutions and Simplifications
- 5.3 Sums and Products
- Notation
- Arithmetic Series
- Geometric Series and the Telescope Tool
- Infinite Series
- 5.4 Polynomials
- Polynomial Operations
- The Zeros of a Polynomial
- 5.5 Inequalities
- Fundamentalldeas
- The AM-GM Inequality
- Massage, Cauchy-Schwarz, and Chebyshev
- 3.4 Invariants
- Chapter
- Chapter
- Chapter
- Combinatorics
- 6.1 Introduction to Counting
- Permutations and Combinations
- Combinatorial Arguments
- Pascal's Triangle and the Binomial Theorem
- Strategies and Tactics of Counting
- 6.2 Partitions and Bijections
- Counting Subsets
- Information Management
- Balls in Urns and Other Classic Encodings
- 6.3 The Principle of Inclusion-Exclusion
- Count the Complement
- PIE with Sets
- PIE with Indicator Functions
- 6.4 Recurrence
- Tiling and the Fibonacci Recurrence
- The Catalan Recurrence
- Number Theory
- 7.1 Primes and Divisibility
- The Fundamental Theorem of Arithmetic
- GCD, LCM, and the Division Algorithm
- 7.2 Congruence
- What's So Good About Primes?
- Fermat's Little Theorem
- 7.3 Number Theoretic Functions
- Divisor Sums
- Phi and Mu
- 7.4 Diophantine Equations
- General Strategy and Tactics
- 7.5 Miscellaneous Instructive Examples
- Can a Polynomial Always Output Primes?
- If You Can Count It, It's an Integer
- A Combinatorial Proof of Fermat's Little Theorem
- Sums of Two Squares
- Geometry for Americans
- 8.1 Three "Easy" Problems
- 8.2 Survival Geometry I
- Points, Lines, Angles, and Triangles
- Chapter xviii CONTENTS
- Parallel Lines
- Circles and Angles
- Circles and Triangles
- 8.3 Survival Geometry II
- Area
- Similar Triangles
- Solutions to the Three "Easy" Problems
- 8.4 The Power of Elementary Geometry
- Concyclic Points
- Area, Cevians, and Concurrent Lines
- Similar Triangles and Collinear Points
- Phantom Points and Concurrent Lines
- 8.5 Transformations
- Symmetry Revisited
- Rigid Motions and Vectors
- Homothety
- Inversion
- Symmetry Revisited
- Calculus
- 9.1 The Fundamental Theorem of Calculus
- 9.2 Convergence and Continuity
- Convergence
- Continuity
- Uniform Continuity
- 9.3 Differentiation and Integration
- Approximation and Curve Sketching
- The Mean Value Theorem
- A Useful Tool
- Integration
- Symmetry and Transformations
- 9.4 Power Series and Eulerian Mathematics
- Don't Worry!
- Taylor Series with Remainder
- Eulerian Mathematics
- Beauty, Simplicity, and Symmetry: The Quest for a Moving Curtain
- References and Further Reading
- Index
ann
(Ann)
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