A Primer for Mathematics Competitions

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The importance of mathematics competitions has been widely recognized for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded.

Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in the final toolchest.

An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement.

As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum.

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Product Details

  • ISBN-13: 9780199539871
  • Publisher: Oxford University Press
  • Publication date: 1/5/2009
  • Pages: 352
  • Product dimensions: 6.40 (w) x 9.30 (h) x 1.10 (d)

Meet the Author

Alexander Zawaira was born in Zimbabwe in 1978. He studied Mathematics and Biochemistry at the University of Zimbabwe where Dr Gavin Hitchcock was one of his teachers. He won a Beit Trust Scholarship to study at Oxford University (England) where he obtained a PhD in Structural Biology. His research interests focus on bridging the gap between bioinformatics and "wet-lab" biochemistry by deriving and experimentally investigating hypotheses from bioinformatics analyses. He is also interested in the general application of mathematics in biology.
Gavin Hitchcock was born in Zimbabwe in 1946. He won scholarships to study mathematics at the Universities of Oxford and Keele, where he took his PhD with a thesis in general topology. He is Senior Lecturer in the Department of Mathematics, University of Zimbabwe, and his research interests are in topology and the history of mathematics. He is internationally known for his writings concerned with the communication of mathematical ideas and their history through theatre and dialogue. He spearheads the mathematical talent search and mathematical Olympiad training programmes in Zimbabwe, and is editor of Zimaths Magazine. He mounts workshops in Zimbabwe and neighbouring countries for teachers, for learners, and for parents, on such topics as "touching and seeing mathematics", "using the history of mathematics to enliven teaching", and "creative problem solving". He also conducts seminars on creative problem solving for workers and for management in commerce and industry.

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Table of Contents

Preface ix

1 Geometry 1

1.1 Brief reminder of basic geometry 3

1.1.1 Geometry of straight lines 4

1.1.2 Geometry of polygons 6

1.1.3 Geometry of the fundamental polygon - the triangle 8

1.1.4 Geometry of circles and circular arcs 13

1.2 Advanced geometry of the triangle 23

1.3 Advanced circle geometry 45

1.4 Problems 54

1.5 Solutions 65

2 Algebraic inequalities and mathematical induction 89

2.1 The method of induction 90

2.2 Elementary inequalities 99

2.3 Harder inequalities 101

2.4 The discriminant of a quadratic expression 110

2.5 The modulus function 112

2.6 Problems 118

2.7 Solutions 120

3 Diophantine equations 125

3.1 Introduction 125

3.2 Division algorithm and greatest common divisor 126

3.3 Euclidean algorithm 127

3.4 Linear Diophantine equations 128

3.4.1 Finding a particular solution of ax + by = c 128

3.4.2 Finding the general solution of ax + by = c 130

3.5 Euclidean reduction, or 'divide and conquer' 131

3.6 Some simple nonlinear Diophantine equations 135

3.7 Problems 138

3.8 Solutions 139

4 Number theory 145

4.1 Divisibility, primes and factorization 146

4.2 Tests for divisibility 147

4.3 The congruence notation: finding remainders 148

4.4 Residue classes 152

4.5 Two useful theorems 154

4.6 The number of zeros at the end of n! 159

4.7 The Unique Factorization Theorem 161

4.8 The Chinese Remainder Theorem 162

4.9 Problems 166

4.10 Solutions 169

5 Trigonometry 181

5.1 Angles and their measurement 182

5.2 Trigonometric functions of acute angles 186

5.3 Trigonometric functions of general angles 189

5.4 Graphs of sine and cosine functions 194

5.5 Trigonometric identities 197

5.5.1 ThePythagorean set of identities 197

5.5.2 Addition formulas 198

5.5.3 Double angle formulas 200

5.5.4 Product formulas 201

5.5.5 Sum formulas 202

5.6 Trigonometric equations 202

5.7 Problems 207

5.8 Solutions 208

6 Sequences and Series 213

6.1 General sequences 214

6.2 The summation notation 214

6.3 Arithmetic Progressions 216

6.4 Geometric Progressions 218

6.5 Sum to infinity of a Geometric Progression 220

6.6 Formulas for sums of squares and cubes 222

6.7 Problems 226

6.8 Solutions 228

7 Binomial Theorem 235

7.1 Pascal's triangle 236

7.2 A formula for the coefficients 239

7.3 Some properties of Pascal's triangle 242

7.4 Problems 244

7.5 Solutions 246

8 Combinatorics (counting techniques) 251

8.1 The fundamental principle of enumeration 252

8.2 Factorial arithmetic 254

8.3 Partitions and permutations of a set 256

8.3.1 Definition of terms 256

8.3.2 The general partitioning formula 256

8.3.3 The general permutation formula 258

8.3.4 Circular permutations 259

8.4 Combinations 265

8.5 Derangements 268

8.6 The exclusion-inclusion principle 272

8.7 The pigeon-hole principle 280

8.8 Problems 283

8.9 Solutions 285

9 Miscellaneous problems and solutions 293

9.1 Problems 293

9.2 Solutions 306

Further Training Resources 337

Index 339

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