Table Of Content_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
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PETERBOROUGH, ONTAR
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Oxford University Press, Great Clarendon Street, Oxford 0X2 6DP
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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
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This book is sold subject to the condition that it shall not,
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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
