內容大鋼
普特南數學競賽（The William Lowell Putnam Mathematical Competition）創辦于1927年，是世界上最負盛名的大學生數學競賽。普特南競賽的桂冠獲獎者（Putnam Fellows）中包括後來的菲爾茲獎得主約翰·米爾諾（John Milnor）、戴維·芒福德（David Mumford）、丹尼爾·奎倫（Daniel Quillen）和諾貝爾物理學獎獲得者理查德·費曼（Richard Feynman）、肯尼斯·威爾遜（Kenneth Wilson）等。
通過嘗試解答普特南數學競賽試題，讀者可以逐漸將其中學層面數學的解題思維轉變為高等數學的解題思維，直至做數學研究的思維方式。本書中也包括不少國際數學奧林匹克競賽（IMO）的試題，以及各國選撥奧數國家隊的賽題，這些題目背後往往有著更為深刻的數學背景。本書以一種循序漸進的方式幫助學習者提升自己的數學能力。
目錄
1 Methods of Proof
1.1 Argumentby Contladiction
1.2 Mathematical Induction
1.3 ThePigeonholePrirciple
1.4 Ordered Sets and Extremal Elements
1.5 Invariants and Semiqnvariants
2 Algebra
2.1 Identities and Inequalities
2.1.1 Algebraic Identities
2.1.2 x2?0
2.1.3 The Cauchy-Schwarz Inequality
2.1.4 The Triangle Inequality
2.1.5 The Arithmetic Mean-Geometric Mean Inequality
2.1.6 Sturm's Principle
2.1.7 Other Inequalities
2.2 Polynomials
2.2.1 A Warmup in One-Variable Polynomials
2.2.2 Polynomials in Several Variables
2.2.3 Quadratic Polynomials
2.2.4 Vi&e's Relations
2.2.5 The Derivative czf a Polynomial
2.2.6 The Location of the Zeros czf a Polynomial
2.2.7 Irreducible Polynomials
2.2.8 Chebyshev Polynomials
2.3 Linear Algebra
2.3.1 Operations with Matrices
2.3.2 Determinants
2.3.3 The Inverse of a Matrix
2.3.4 Systems of Linear Equations
2.3.5 Vector Spaces, Linear Combinations of Vectors, Bases
2.3.6 Linear Transformations, Eigenvalues, Eigenvectors
2.3.7 The Cayley-Hamilton and Perron-Frobenius Theorems
2.4 Abstract Algebra
2.4.1 Binary Operations
2.4.2 Groups
2.4.3 Rings
3 Real Analysis
3.1 Sequences and Series
3.1.1 Search for a Pattern
3.1.2 Linear Recursive Sequences
3.1.3 Limits of Sequences
3.1.4 More About Limits of Sequences
3.1.5 Series
3.1.6 Telescopic Series and Products
3.2 Continuity, Derivatives, and Integrals
3.2.1 Functions
3.2.2 Limits of Functions
3.2.3 Continuous Functions
3.2.4 The Intermediate Value Property
3.2.5 Derivatives and Their Applications
3.2.6 The Mean Value Theorem
3.2.7 Convex Functions
3.2.8 Indefinite Integrals
3.2.9 Definite Integrals
3.2.10 Riemann Sums
3.2.11 Inequalities for Integrals
3.2.12 Taylor and Fourier Series
3.3 Multivariable Differential and Integral Calculus
3.3.1 Partial Derivatives and Their Applications
3.3.2 Multivariable Integrals
3.3.3 The Many Versions of Stokes' Theorem
3.4 Equations with Functions as Unknowns
3.4.1 Functional Equations
3.4.2 Ordinary Differential Equations of the First Order
3.4.3 Ordinary Differential Equations of Higher Order
3.4.4 Problems Solved with Techniques of Differential Equations
4 Geometry and Trigonometry
4.1 Geometry
4.1.1 Vectors
4.1.2 The Coordinate Geometry of Lines and Circles
4.1.3 Quadratic and Cubic Curves in the Plane
4.1.4 Some Famous Curves in the Plane
4.1.5 Coordinate Geometry in Three and More Dimensions
4.1.6 Integrals in Geometry
4.1.7 Other Geometry Problems
4.2 Trigonometry
4.2.1 Trigonometric Identities
4.2.2 Euler's Formula
4.2.3 Trigonometric Substitutions
4.2.4 Telescopic Sums and Products in Trigonometry
5 Number Theory
5.1 Integer-Valued Sequences and Functions
5.1.1 Some General Problems
5.1.2 Fermat's Infinite Descent Principle
5.1.3 The Greatest Integer Function
5.2 Arithmetic
5.2.1 Factorization and Divisibility
5.2.2 Prime Numbers
5.2.3 Mcdular Arithmetic
5.2.4 Fermat's Little Theorem
5.2.5 Wilson's Theorem
5.2.6 Euler's Totient Function
5.2.7 The Chinese Remainder Theorem
5.2.8 Quadratic IntcgerRings
5.3 Diophantine Equations
5.3.1 Linear Dicphantine Equations
5.3.2 The Equation of Pythagoras
5.3.3 Pell's Equation
5.3.4 Other Diophantine Equations
6 Combinatories and Probability
6.1 Combinatorial Arguments in Set Theory
6.1.1 Combinatorics of Sets
6.1.2 Combinatorics of Numbers
6.1.3 Permutations
6.2 Combinatorial Geometry
6.2.1 Tessellations
6.2.2 Miscellaneous Combinatorial Geometry Problems
6.3 Graphs
6.3.1 Some Basic Graph Theory
6.3.2 Euler's Formula for Planar Graphs
6.3.3 Ramsey Theory
6.4 Binomial Coefficients and Counting Methods
6.4.1 Combinatorial Identities
6.4.2 Generating Functions
6.4.3 Counting Strategies
6.4.4 The Inclusion-Exclusion Principle
6.5 Probability
6.5.1 Equally Likely Cases
6.5.2 Establishing Relaticns Among Probabilities
6.5.3 Geometric Probabilities
Methods of Proof
Algebra
Real Analysis
Geometry and Trigonometry
Number Theory
Combinatorics and Probability
Index of Notation
Index
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