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Advanced book on Mathematics Olympiad

(ff) #1

  • 1 Methods of Proof................................................ A Study Guide...................................................... xv

    • 1.1 Argument by Contradiction......................................

    • 1.2 Mathematical Induction.........................................

    • 1.3 The Pigeonhole Principle........................................

    • 1.4 Ordered Sets and Extremal Elements..............................

    • 1.5 Invariants and Semi-Invariants...................................



  • 2 Algebra........................................................

    • 2.1 Identities and Inequalities

      • 2.1.1 Algebraic Identities.......................................

      • 2.1.2 x^2 ≥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

      • 2.2.2 Viète’s Relations.........................................

      • 2.2.3 The Derivative of a Polynomial.............................

      • 2.2.4 The Location of the Zeros of a Polynomial...................

      • 2.2.5 Irreducible Polynomials...................................

      • 2.2.6 Chebyshev Polynomials...................................



    • 2.3 Linear Algebra................................................. viii Contents

      • 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 Limits of Functions.......................................

      • 3.2.2 Continuous Functions.....................................

      • 3.2.3 The Intermediate Value Property............................

      • 3.2.4 Derivatives and Their Applications..........................

      • 3.2.5 The Mean Value Theorem

      • 3.2.6 Convex Functions........................................

      • 3.2.7 Indefinite Integrals

      • 3.2.8 Definite Integrals.........................................

      • 3.2.9 Riemann Sums

      • 3.2.10 Inequalities for Integrals...................................

      • 3.2.11 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.............. Contents ix

      • 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 Conics and Other Curves in the Plane........................

      • 4.1.4 Coordinate Geometry in Three and More Dimensions..........

      • 4.1.5 Integrals in Geometry.....................................

      • 4.1.6 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 Modular 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...........................



    • 5.3 Diophantine Equations..........................................

      • 5.3.1 Linear Diophantine Equations..............................

      • 5.3.2 The Equation of Pythagoras................................

      • 5.3.3 Pell’s Equation

      • 5.3.4 Other Diophantine Equations...............................





  • 6 Combinatorics and Probability....................................

    • 6.1 Combinatorial Arguments in Set Theory and Geometry...............

      • 6.1.1 Set Theory and Combinatorics of Sets.......................

      • 6.1.2 Permutations............................................

      • 6.1.3 Combinatorial Geometry..................................

        • 6.1.4 Euler’s Formula for Planar Graphs.......................... x Contents

        • 6.1.5 Ramsey Theory..........................................



      • 6.2 Binomial Coefficients and Counting Methods.......................

        • 6.2.1 Combinatorial Identities...................................

        • 6.2.2 Generating Functions.....................................

        • 6.2.3 Counting Strategies.......................................

        • 6.2.4 The Inclusion–Exclusion Principle..........................



      • 6.3 Probability....................................................

        • 6.3.1 Equally Likely Cases.....................................

        • 6.3.2 Establishing Relations Among Probabilities

        • 6.3.3 Geometric Probabilities...................................





    • Methods of Proof.................................................... Solutions



  • Algebra............................................................

    • Real Analysis.......................................................

    • Geometry and Trigonometry..........................................

    • Number Theory.....................................................

    • Combinatorics and Probability........................................

    • Index of Notation

    • Index..............................................................



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