_OXFORD SCIENCE PUBLICATIONS THE MATHEMATICAL OLYMPIAD HANDBOOK AN INTRODUCTION TO The Mathematical Olympiad Handbook «■ The Mathematical Olympiad Handbook An introduction to problem solving based on the first 32 British Mathematical Olympiads 1965-1996 A. GARDINER School of Mathematics University of Birmingham, UK Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1997 1 REN i uNi VtiOi * PETERBOROUGH, ONTAR 3 ci'5 /<?97 Oxford University Press, Great Clarendon Street, Oxford 0X2 6DP Oxford New York Athens Auckland Bangkok Bogota Bombay Buenos Aires Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press Inc., New York © A. Gardiner, 1997 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Gardiner, A. (Anthony), 1947- The Mathematical Olympiad Handbook: an introduction to problem solving based on the first 32 British mathematical olympiads 1965-1996 / A. Gardiner. 1. Mathematics-Problems, exercises, etc. 2. British Mathematical Olympiad. I. Title. QA43.G345 1997 510'.76~dc21 97-17647 ISBN 0 19 850105 6 Typeset by Technical Typesetting Ireland Printed in Great Britain by Biddles Ltd., Guildford & King’s Lynn ... a mathematical problem should be difficult in order to entice us, yet not completely inaccessible lest it mock our efforts. It should be a guidepost on the tortuous path to hidden truths, ultimately rewarding us by the satisfaction of success in its solution. David Hilbert, 1900 Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation https://archive.org/details/mathematicalolym1997gard Preface The supply of problems in mathematics is inexhaustible; no sooner is one problem solved than numerous others spring forth in its place. David Hilbert, 1990 This is unashamedly a book for beginners. Unlike most Olympiad problem books, my aim has been to convince as many people as possible that Mathematical Olympiad problems are for them and not just for some bunch of freaks. I have tried never to use one slick, unmotivated step where three down-to-earth steps would do just as well. Once you are convinced that these problems are do-able and are worth the effort required to solve them, you will be ready to move on to other books of Olympiad problems—some of which are listed in the section ‘Some books for your bookshelf. Who is the book for? In mathematics, as in music and sport, there are many youngsters who are capable of performing at a much higher level than is required by the ordinary school curriculum. While many youngsters have the necessary potential, the final ability is not God-given: it has to be developed—and that requires effort and commitment. Such students may remain in their peer groups, but need higher goals to aim at. This book is for such students aged 15 or 16 and above. The book presumes an initial commitment on the part of you—the reader. At first, this commitment need go no further than a basic technical compe¬ tence, combined with a willingness to struggle on when faced with a challeng¬ ing mathematical problem. But sooner or later you will have to make the effort to follow up the mathematical ideas which lie behind these problems, and start reading other books to master some unfamiliar mathematics on your own. Reader-friendly mathematics books can be hard to find: to guide you, the section ‘Some books for your bookshelf contains a short list of books which you may find helpful, together with a very brief summary of the mathematics covered by each one. What is in this book? The main part of this book contains problems from the first 32 British Mathematical Olympiad (BMO) papers 1965-96, starting with 1996 and Preface viii working backwards. This is followed by a section in which the problems are considered in turn, with each problem being followed by a page or so of text. The first part of this text should be read only after you have had a good go at the problem for yourself. You should then use these fresh ideas for a second attack on the problem. Only later—whether or not you have succeeded in solving the problem—should you work through the whole ‘Outline solution’. If you failed to solve the problem, the text should convince you that there is at least one ‘obvious’ line of attack which you could have discovered for yourself. If on the other hand you managed to solve the problem on your own, the text may still open your eyes to some things you never thought of. However, the text itself does not present a complete solution. To obtain a complete solution you must fill in for yourself the details in the ‘obvious’ approach outlined in the text. Most of the problems in this book can be solved in many different ways. The first correct solution that one finds to a problem is almost never the simplest. It is only after one has succeeded in solving a problem that one can sometimes look back and see that there is in fact a much easier solution—if only one looks at things in the right way! The papers from the years 1965-74 are different in style, in that each contains ten or eleven (not necessarily original) problems—including the occasional mechanics problem. I have included the statements of the prob¬ lems both for completeness and because they include many nice examples; however, I have not provided any discussion, hints, or outline solutions for these problems. Most Olympiad problems do not require advanced mathematics for their solution. However, they do require familiarity with some basic ideas which receive little attention in most secondary school curricula. The section ‘A little useful mathematics’ provides a brief introduction to the simplest and most basic topics. You may find it helpful to work through this section for a first time before tackling the problems in the rest of the book. Later, as you meet problems that use some of these ideas, you may need to work through some parts of the section in more detail. Why was the book written? Textbook exercises and examination questions are not meant to be particu¬ larly demanding: textbook exercises are intended to help all students practise some standard technique, and exam questions are supposed to give ordinary students an opportunity to show what they have learned. Thus both these types of problems should be relatively straightforward for the large number of students who are capable of performing at a higher level. This does not mean they should be skipped! Indeed, it is perhaps more important for able