A First Step to Mathematical Olympiad Problems
Mathematical Olympiad Series ISSN: 1793-8570 Series Editors: Lee Peng Yee (Nanyang Technological University, Singapore) Xiong Bin (East China Normal University, China) A First Step to Mathematical Olympiad Problems Downloaded from www.worldscientific.com Vol. 1 A First Step to Mathematical Olympiad Problems by Derek Holton (University of Otago, New Zealand)
Derek Holton University of Otago, New Zealand A First Step to Mathematical Olympiad Problems Downloaded from www.worldscientific.com Vol. 1 Mathematical Olympiad Series A First Step to Mathematical Olympiad Problems World Scientific N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE A First Step to Mathematical Olympiad Problems Downloaded from www.worldscientific.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. A FIRST STEP TO MATHEMATICAL OLYMPIAD PROBLEMS Mathematical Olympiad Series Vol. 1 Copyright 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4273-87-9 (pbk) ISBN-10 981-4273-87-2 (pbk) Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore.
To Marilyn, for all her help and encouragement
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Foreword The material in this book was first written for students in New Zealand who were preparing to compete for the six positions in New Zealand s International Mathematical Olympiad (IMO) team. At that stage there was very little mathematical writing available for students who were good at high school mathematics but not yet competent to tackle IMO problems. The aim of the material here then was to give those students sufficient background in areas of mathematics that are commonly the subject of IMO questions so that they were ready for IMO standard work. This book covers discrete mathematics, number theory and geometry with a final chapter on some IMO problems. So this book can provide a basis for the initial training of potential IMO students, either with students in a group or for students by themselves. However, I take the approach that solving problems is what mathematics is all about and my second aim is to introduce the reader to what I believe is the essence of mathematics. In many classrooms in many countries, mathematics is presented as a collection of techniques that have to be learnt, often just to be reproduced in examinations. Here I try to present the other, creative, side of the mathematical coin. This is a side that I believe to be far more interesting and exciting. It is also the side that enables students to get some idea of the way that research mathematicians approach their work. So this book can be used to start students on the trail towards the IMO but its broader aim is to start students on a trail to understanding what mathematics really is and then possibly to taking that understanding and using it in later life, both inside mathematics and outside it. I would like to thank Irene Goodwin, Leanne Kirk, Lenette Grant, Lee Peng Yee and Zhang Ji for all of their assistance in the preparation of this book. vii
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Foreword Contents 1. Jugs and Stamps: How To Solve Problems 1 1.1. Introduction... 1 1.2. A Drinking Problem.................... 2 1.3. About Solving Problems................. 3 1.4. Rethinking Drinking.................... 5 1.5. Summing It Up...................... 7 1.6. Licking a Stamp Problem................. 10 1.7. A Little Explanation................... 12 1.8. Tidying Up......................... 12 1.9. Generalise......................... 14 1.10. In Conclusion....................... 16 1.11. Epilogue.......................... 18 1.12. Solutions.......................... 19 2. Combinatorics I 27 2.1. Introduction... 27 2.2. What is Combinatorics?.................. 27 2.3. The Pigeonhole Principle................. 29 2.4. Counting without Counting................ 36 2.5. A Sigma Aside....................... 48 2.6. Solutions.......................... 49 vii 3. Graph Theory 61 3.1. Introduction... 61 3.2. Königsberg......................... 61 ix
x A First Step to Mathematical Olympiad Problems A First Step to Mathematical Olympiad Problems Downloaded from www.worldscientific.com 3.3. SoWhatisaGraph?... 65 3.4. Ramsey... 72 3.5. Euler Tours (Revisited).................. 74 3.6. Knight s Tours....................... 75 3.7. Hamilton.......................... 78 3.8. Trees............................ 80 3.9. Planarity.......................... 81 3.10. The Four Colour Theorem................ 85 3.11. Some Additional Problems................ 91 3.12. Solutions... 93 4. Number Theory 1 108 4.1. What is It?......................... 108 4.2. Divisibility by Small Numbers.............. 110 4.3. Common Factors...................... 116 4.4. Fermat s Little Theorem................. 124 4.5. A.P. s............................ 127 4.6. Some More Problems................... 131 4.7. Solutions.......................... 132 5. Geometry 1 146 5.1. Introduction........................ 146 5.2. Squares........................... 146 5.3. Rectangles and Parallelograms.............. 151 5.4. Triangles.......................... 153 5.5. Circles........................... 157 5.6. Solutions.......................... 162 6. Proof 180 6.1. Introduction........................ 180 6.2. Why Proof?........................ 180 6.3. Proof by Contradiction.................. 184 6.4. Mathematical Induction.................. 186 6.5. Conclusion......................... 194 6.6. Solutions.......................... 195
Contents xi A First Step to Mathematical Olympiad Problems Downloaded from www.worldscientific.com 7. Geometry 2 208 7.1. Cartesian Geometry.................... 208 7.2. Lines............................ 208 7.3. Modulus.......................... 214 7.4. Loci: One Fixed Point................... 218 7.5. The Cosine Rule...................... 221 7.6. Loci:TwoPoints... 223 7.7. Conics........................... 229 7.8. Solutions.......................... 230 8. Some IMO Problems 249 8.1. Introduction... 249 8.2. What is the IMO?..................... 249 8.3. PHIL 1........................... 250 8.4. MON 1........................... 252 8.5. MON 6........................... 253 8.6. UNK 2........................... 254 8.7. Hints PHIL 1...................... 254 8.8. Hints MON 1...................... 256 8.9. Hints MON6... 258 8.10. Hints UNK2... 259 8.11. Solutions.......................... 261 Index 275