Table Of ContentProblem Books in Mathematics
Edited by P.R. Halmos
Springer
New York
Berlin
Heidelberg
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
Problem Books in Mathematics
Series Editor: P.R. Halmos
Polynomials
by Edward J. Barbeau
PvoblemsinGeome~
by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint
Raymond
Pvoblem Book for First Year Calculus
by George W Bluman
Exercises in Pvobability
by T CacouZZos
An Intvoduction to Hilbert Space and Quantum Logic
by David W Cohen
Unsolved Pvoblems in Geome~
by HaZZard T Croft, Kenneth J. Falconer, and Richard K. Guy
Pvoblems in Analysis
by Bernard R. Gelbaum
Pvoblems in Real and Complex Analysis
by Bernard R. Gelbaum
Theorems and Counterexamples in Mathematics
by Bernard R. Gelbaum and John M.H. Olmsted
Exercises in Integration
by Claude George
Algebraic Logic
by S. G. Gindikin
(continued after index)
Edward Lozansky Cecil Rousseau
Winning Solutions
Springer
Edward Lozansky Cecil Rousseau
National Science Thacher's Association The University of Memphis
Washington, DC 20009 Memphis, TN 38152
USA USA
Series Editor:
Paul R. Halmos
Department of Mathematics
Santa Clara University
Santa Clara, CA 95053
USA
Mathematics Subject Classification (1991): llAxx 05Axx
Library of Congress Cataloging-in-Publication Data
Lozansky, Edward
Winning Solutions! Edward Lozansky, Cecil Rousseau.
p. cm - (Problem books in mathematics)
Includes bibliographical references (p. -) and index.
ISBN-13: 978-0-387-94743-3 e-ISBN-I3: 978-1-4612-4034-1
DOl: 10.1007/978-I -46 I 2-4034-I
1. Mathematics - Problems, exercises, etc. I. Rousseau, Cecil. II. Title. III. Series
QA43.L793 1996
5lO'.76-dc20 96-13584
Printed on acid-free paper.
@ 1996 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
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Production managed by Robert Wexler; manufacturing supervised by Joe Quatela.
Photocomposed copy prepared using the author's IdJ':&C files and Springer's utm macro.
987654321
SPIN 10016809
Preface
Problem-solving competitions for mathematically talented sec
ondary school students have burgeoned in recent years. The number
of countries taking part in the International Mathematical Olympiad
(IMO) has increased dramatically. In the United States, potential
IMO team members are identified through the USA Mathematical
Olympiad (USAMO), and most other participating countries use a
similar selection procedure. Thus the number of such competitions
has grown, and this growth has been accompanied by increased
public interest in the accomplishments of mathematically talented
young people.
There is a significant gap between what most high school math
ematics programs teach and what is expected of an IMO participant.
This book is part of an effort to bridge that gap. It is written for
students who have shown talent in mathematics but lack the back
ground and experience necessary to solve olympiad-level problems.
We try to provide some of that background and experience by point
ing out useful theorems and techniques and by providing a suitable
collection of examples and exercises.
This book covers only a fraction of the topics normally rep
resented in competitions such as the USAMO and IMO. Another
volume would be necessary to cover geometry, and there are other
v
VI Preface
special topics that need to be studied as part of preparation for
olympiad-level competitions. At the end of the book we provide a
list of resources for further study.
A word of explanation is due the reader who is not already fa
miliar with olympiads and the topics normally dealt with in such
competitions. Until now, calculus has not been accepted as one of
those topics. Problems on olympiad exams regularly call for use of
Ceva's theorem, Chebyshev's inequality, the Chinese remainder the
orem, and convex sets, but not calculus. The authors are the first to
acknowledge that this book deals with an ecclectic list of topics.
However, we have tried to choose these topics with the olympiad
tradition and the needs of mathematically talented young persons
in mind.
Many people have made valuable suggestions to us during the
writing of this book. We are especially grateful to Basil Gordon
(UCLA), Ian McGee (University of Waterloo), and Ron Scoins (Uni
versity of Waterloo) for suggestions made concerning the first two
chapters, and to David Dwiggins (University of Memphis) for his
careful reading of the final manuscript.
The first two chapters of this book were written while one of
the authors [CR] was on sabbatical at the University of Waterloo.
This author wishes to thank Ron Dunkley for the invitation to visit
Waterloo and to express his appreciation to all the members of the
faculty and staff who helped make this visit a productive one.
For one of the authors [CR], the opportunity to write this book is
an outgrowth of the good fortune of having been associated with
both the USAMO and the IMO for many years. The opportunity
for this author to play such a role was initially provided by Murray
Klamkin, and has been supported and enlarged by many others, in
cluding Dick Gibbs, Samuel Greitzer, Walter Mientka, Ian Richards,
Leo Schneider, many fine colleagues of the Mathematical Olympiad
Summer Program (Titu Andreescu, Anne Hudson, Gregg Patruno,
Gail Ratcliff, Daniel Ullman, Elizabeth Wilmer), and the many won
derfully talented students who have participated in the USAMO,
IMO, and the Mathematical Olympiad Summer Program.
Finally, we are very grateful to the American Mathematical Com
petitions for permission to use problems from the AIME (American
Preface VB
Invitational Mathematics Examination) and the USAMO as examples
and exercises in this book.
January, 1996
Edward Lozansky Cecil Rousseau
Washington, D.C. Memphis, TN
Contents
Preface v
1 Numbers 1
1.1 The Natural Numbers . 1
1.2 Mathematical Induction 11
1.3 Congruence....... 18
1.4 Rational and Irrational Numbers . 29
1.5 Complex Numbers ... 35
1.6 Progressions and Sums 46
1.7 Diophantine Equations 56
1.8 Quadratic Reciprocity . 65
2 Algebra 73
2.1 Basic Theorems and Techniques 73
2.2 Polynomial Equations ..... . 92
2.3 Algebraic Equations and Inequalities 106
2.4 The Classical Inequalities ..... . 113
3 Combinatorics 141
3.1 What is Combinatorics? . 141
3.2 Basics of Counting .... 142
IX
X Contents
3.3 Recurrence Relations . . . . . . . 149
3.4 Generating Functions . . . . . . . 156
3.5 The Inclusion-Exclusion Principle 178
3.6 The Pigeonhole Principle. 188
3.7 Combinatorial Averaging 195
3.8 Some Extremal Problems . 202
Hints and Answers for Selected Exercises 215
General References 237
List of Symbols 239
Index 241
Numbers
CHAPTER
1.1 The Natural Numbers
Normally, we first learn about mathematics through counting,
so the first set of numbers encountered is the set of counting
numbers or natural numbers {I, 2, 3, ... }. Later, our knowledge is
extended to integers, rational numbers, real numbers and com
plex numbers. A formal definition of even the natural number
system requires careful thought, and one was given only in 1889
by the Italian mathematician Giuseppe Peano. Our approach is in
formal. It is assumed that the reader is familiar with various number
systems. The following definitions ensure a common language with
which to present problems and their solutions.
We use Z to denote the set of integers {. .. , -2, -1,0, 1,2, ... } and
Z+ to signify the set of positive integers {I, 2, 3, ... }. We shall use the
term natural number to mean a positive integer. (Mathematicians
do not always agree on matters of terminology and notation. Some
use the term natural number to mean a nonnegative integer.) If a
and b are integers, we say that a divides b, in symbols alb, if there
is an integer c such that b = ac. Then a is a divisor, or factor, ofb.
A natural number p > 1 is said to be prime if 1 and p are its only
positive divisors. A natural number n > 1 that is not prime is said to
1
