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Mathematical buffet

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This book is a buffet of mathematical methods for problem solving. It ­contains a large collection of problems that have been carefully selected and organized in a way to help preparing for olympiad level competitions. The book also contains a thorough description of the tools and techniques that can be used to solve these problems. Special emphasis is put on ­describing how to think and solve problems like a professional mathematician, giving plenty of hands-on advice that has shown to be ef...

This book is a buffet of mathematical methods for problem solving. It ­contains a large collection of problems that have been carefully selected and organized in a way to help preparing for olympiad level competitions. The book also contains a thorough description of the tools and techniques that can be used to solve these problems. Special emphasis is put on ­describing how to think and solve problems like a professional mathematician, giving plenty of hands-on advice that has shown to be effective. The book is primarily written for students involved in highschool level mathematical competitions, on national as well as international level, but could also be of interest to highschool teachers and university students with an interest in mathematical problem solving.

    • 9
      Introduction
  • PART I Basic ideas and facts
      • 1
        15
        CHAPTER 1 General advice
        • 1.1
          16
          Tiny mathematical plays
        • 1.2
          20
          The importance of make-up
        • 1.3
          25
          Looking for bridges
        • 1.4
          27
          Finding weak points
        • 1.5
          29
          On the other side of the river
        • 1.6
          31
          Complete the picture!
        • 1.7
          33
          Reversing the theorems
        • 1.8
          39
          The art of listening
        • 1.9
          42
          The inertia of thinking
        • 1.10
          47
          To know and to understand
        • 1.11
          50
          To see the difference
        • 1.12
          55
          Finding the invariants
        • 1.13
          57
          Do the opposite!
        • 1.14
          62
          Exercises
        • 1.15
          64
          Hints
      • 2
        65
        CHAPTER 2 The Numbers
        • 2.1
          65
          Natural numbers and integers
        • 2.2
          70
          Arithmetic in Zm
        • 2.3
          75
          Diophantine equations
        • 2.4
          76
          Rational and irrational numbers
        • 2.5
          80
          Real numbers
        • 2.6
          82
          Complex numbers
        • 2.7
          93
          Exercises
        • 2.8
          96
          Hints
      • 3
        97
        CHAPTER 3 Algebra
        • 3.1
          97
          Identities and factorization
        • 3.2
          99
          Polynomials in one variable
        • 3.3
          105
          Symmetric polynomials
        • 3.4
          108
          Summing up
        • 3.5
          112
          Inequalities: basic properties
        • 3.6
          121
          Proving inequalities
        • 3.7
          135
          Exercises
        • 3.8
          138
          Hints
      • 4
        139
        CHAPTER 4 Geometry
        • 4.1
          139
          The anatomy of a triangle
        • 4.2
          145
          The degree of freedom
        • 4.3
          151
          How to remember trigonometry
        • 4.4
          157
          Angle chasing
        • 4.5
          168
          Centre of gravity and barycentric coordinates
        • 4.6
          175
          When are the points collinear?
        • 4.7
          182
          Equal sizes in a triangle
        • 4.8
          189
          Exercises
        • 4.9
          190
          Hints
      • 5
        191
        CHAPTER 5 Geometric transformations
        • 5.1
          191
          Reflection
        • 5.2
          196
          Homothety
        • 5.3
          200
          Exercises
        • 5.4
          202
          Hints
      • 6
        203
        CHAPTER 6 Combinatorics
        • 6.1
          203
          Enumerating the objects
        • 6.2
          207
          Elements of graph theory
        • 6.3
          212
          The pigeonhole principle
        • 6.4
          216
          Colouring the problem
        • 6.5
          216
          Exercises
        • 6.6
          218
          Hints
      • 7
        219
        CHAPTER 7 Functions and functional equations
        • 7.1
          219
          Functions and permutations
        • 7.2
          226
          Functional equations
        • 7.3
          233
          Exercises
        • 7.4
          234
          Hints
  • PART II Advanced tools
      • 8
        237
        CHAPTER 8 Number Theory
        • 8.1
          237
          Order and primitive elements
        • 8.2
          239
          Quadratic equations in Z p and the Legendre symbol
        • 8.3
          242
          Continuous fractions and Pell’s equation
      • 9
        249
        CHAPTER 9 Elements of linear algebra
        • 9.1
          249
          Matrices
        • 9.2
          259
          Vector algebra
        • 9.3
          262
          Real numbers as a rational vector space
        • 9.4
          264
          Finite-dimensional vector spaces inside of R
      • 10
        269
        CHAPTER 10 Real functions
        • 10.1
          269
          Lagrange’s Mean Value Theorem
        • 10.2
          270
          Some elements of topology
        • 10.3
          274
          Completeness and Baire’s theorem
        • 10.4
          277
          Fixpoints
        • 10.5
          280
          Trigonometric polynomials
        • 10.6
          282
          Infinite sums
      • 11
        285
        CHAPTER 11 Algebraic constructions
        • 11.1
          285
          Formal series
        • 11.2
          290
          Combinatorics on words
        • 11.3
          294
          More about sequences
      • 12
        297
        CHAPTER 12 Advanced geometry
        • 12.1
          297
          Identities and inequalities in triangles
        • 12.2
          300
          Directed angles
        • 12.3
          302
          Inversion
        • 12.4
          311
          Affine and projective maps
        • 12.5
          317
          Geometry in Figures
      • 13
        319
        CHAPTER 13 Combinatorial arguments
        • 13.1
          319
          König’s lemma
        • 13.2
          320
          Ramsey’s theorem
        • 13.3
          325
          Playing games
        • 13.4
          328
          Combinatorial geometry
  • PART III Problems
      • 14
        335
        CHAPTER 14 Papers
        • 14.1
          335
          Paper 1
        • 14.2
          336
          Paper 2
        • 14.3
          337
          Paper 3
        • 14.4
          338
          Paper 4
        • 14.5
          339
          Paper 5
        • 14.6
          340
          Paper 6
        • 14.7
          341
          Paper 7
        • 14.8
          342
          Paper 8
        • 14.9
          343
          Paper 9
        • 14.10
          344
          Paper 10
        • 14.11
          345
          Paper 11
        • 14.12
          346
          Paper 12
        • 14.13
          347
          Paper 13
        • 14.14
          348
          Paper 14
        • 14.15
          349
          Paper 15
        • 14.16
          350
          Paper 16
        • 14.17
          351
          Paper 17
        • 14.18
          352
          Paper 18
        • 14.19
          353
          Paper 19
        • 14.20
          354
          Paper 20
  • PART IV Solutions
      • 15
        357
        CHAPTER 15 Solutions to all papers
      • 16
        487
        CHAPTER 16 Sources
    • 489
      Bibliography
    • 493
      Index

Information

Författare:
Victor Ufnarovski Jana Madjarova Frank Wikström
Språk:
Engelska
ISBN:
9789144111032
Utgivningsår:
2016
Artikelnummer:
39120-01
Upplaga:
Första
Sidantal:
498

Författare

Victor Ufnarovski

Victor Ufnarovski was educated at Moscow University. He is a professor of Mathematics at Lund University. His main research is Algebra, in particul...

Jana Madjarova

Jana Madjarova is a professor of mathematics at Chalmers University of Technology/University of Gothenburg She is the chair of the Swedish Mathemat...

Frank Wikström

Frank Wikström är docent i matematik vid Lunds universitet. Han har tidigare även undervisat vid Umeå universitet och Mittuniversitetet och arbetat...

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