Oranges are not the only Fruit

The International Mathematical Olympiad (Imo) is an annual international mathematics competition held for pre-collegiate students. It is also the oldest of the international science olympiads, and competition for places is particularly fierce. This book is an amalgamation of the booklets originally produced to guide students intending to contend for placement on their country's Imo team. See also A First Step to Mathematical Olympiad Problems which was published in 2009. The material contained in this book provides an introduction to the main mathematical topics covered in the Imo, which Combinatorics, Geometry and Number Theory. In addition, there is a special emphasis on how to approach unseen questions in Mathematics, and model the writing of proofs. Full answers are given to all questions. Though A Second Step to Mathematical Olympiad Problems is written from the perspective of a mathematician, it is written in a way that makes it easily comprehensible to adolescents. This book is also a must-read for coaches and instructors of mathematical competitions. Contents 10 Foreword 8 1. Combinatorics 14 1.1. A Quick Reminder 14 1.2. Partial Fraction 17 1.3. Geometric Progressions 22 1.4. Extending the Binomial Theorem 25 1.5. Recurrence Relations 26 1.6. Generating Functions 33 1.7. Of Rabbits and Postmen 37 1.8. Solutions 42 2. Geometry 3 54 2.1. The Circumcircle 55 2.2. Incircles 59 2.3. Exercises 62 2.4. The 6-Point Circle? 64 2.5. The Euler Line and the Nine Point Circle 69 2.6. Some More Examples 71 2.7. Hints 73 2.8. Solutions 78 2.9. Glossary 97 3. Solving Problems 98 3.1. Introduction 98 3.2. A Problem to Solve 98 3.3. Mathematics: What is it? 100 3.4. Back to Six Circles 103 3.5. More on Research Methods 104 3.6. Georg P lya 111 3.7. Asking Questions 113 3.8. Solutions 115 4. Number Theory 2 120 4.1. A Problem 120 4.2. Euler’s φ-function 123 4.3. Back to Section 4.1 125 4.4. Wilson 128 4.5. Some More Problems 130 4.6. Solutions 131 5. Means and Inequalities 146 5.1. Introduction 146 5.2. Rules to Order the Reals By 146 5.3. Means Arithmetic and Geometric 148 5.4. More Means 152 5.5. More Inequalities 155 5.6. A Collection of Problems 159 5.7. Solutions 160 6. Combinatorics 3 180 6.1. Introduction 180 6.2. Inclusion–Exclusion 180 6.3. Derangements (Revisited) 185 6.4. Linear Diophantine Equations Again 187 6.5. Non-taking Rooks 189 6.6. The Board of Forbidden Positions 196 6.7. Stirling Numbers 200 6.8. Some Other Numbers 202 6.9. Solutions 207 7. Creating Problems 228 7.1. Introduction 228 7.2. Counting 229 7.3. Packing 233 7.4. Intersecting 237 7.5. Chessboards 243 7.6. Squigonometry 246 7.7. The Equations of Squares 249 7.8. Solutions 251 8. IMO Problems 2 270 8.1. Introduction 270 8.2. AUS 3 271 8.3. HEL 2 271 8.4. TUR 4 272 8.5. ROM 4 272 8.6. USS 1 273 8.7. Revue 273 8.8. Hints — AUS 3 274 8.9. Hints — HEL 2 276 8.10. Hints — TUR 4 277 8.11. Hints — ROM 4 279 8.12. Hints — USS 1 281 8.13. Some More Olympiad Problems 282 8.14. Solutions 285 Index 308 Combinatorics 2 Geometry 3 Solving Problems Number Theory 2 Inequalities Combinatorics 3 IMO Problems 2 Creating Problems.