USA & International Mathematical Olympiads 2003

101 Pages·2004·4.214 MB·English

by  Titu Andreescu, Zuming Feng

USA and International
Mathematical Olympiads
2003
c 2004 by
°
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2004?????
ISBN 0-88385-???-?
Printed in the United States of America
Current Printing (last digit):
10 9 8 7 6 5 4 3 2 1
USA and International
Mathematical Olympiads
2003
Edited by
Titu Andreescu
and
Zuming Feng
Published and distributed by
The Mathematical Association of America
Council on Publications
Roger Nelsen, Chair
Roger Nelsen Editor
Irl Bivens Clayton Dodge
Richard Gibbs George Gilbert
Gerald Heuer Elgin Johnston
Kiran Kedlaya Loren Larson
Margaret Robinson Mark Saul
MAAPROBLEMBOOKSSERIES
ProblemBooksisaseriesoftheMathematicalAssociationofAmericaconsisting
of collections of problems and solutions from annual mathematical competitions;
compilations of problems (including unsolved problems) specific to particular
branchesofmathematics;booksontheartandpracticeofproblemsolving,etc.
AFriendlyMathematicsCompetition:35YearsofTeamworkinIndiana,editedby
RickGillman
The Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and Loren C.
Larson
InternationalMathematicalOlympiads1986–1999,MarcinE.Kuczma
Mathematical Olympiads 1998–1999: Problems and Solutions From Around the
World,editedbyTituAndreescuandZumingFeng
Mathematical Olympiads 1999–2000: Problems and Solutions From Around the
World,editedbyTituAndreescuandZumingFeng
Mathematical Olympiads 2000–2001: Problems and Solutions From Around the
World,editedbyTituAndreescu,ZumingFeng,andGeorgeLee,Jr.
The William Lowell Putnam Mathematical Competition Problems and Solutions:
1938–1964,A.M.Gleason,R.E.Greenwood,L.M.Kelly
The William Lowell Putnam Mathematical Competition Problems and Solutions:
1965–1984,GeraldL.Alexanderson,LeonardF.Klosinski,andLorenC.Larson
The William Lowell Putnam Mathematical Competition 1985–2000: Problems,
Solutions,andCommentary, KiranS.Kedlaya,BjornPoonen,RaviVakil
USA and International Mathematical Olympiads 2000, edited byTitu Andreescu
andZumingFeng
USA and International Mathematical Olympiads 2001, edited byTitu Andreescu
andZumingFeng
USA and International Mathematical Olympiads 2002, edited byTitu Andreescu
andZumingFeng
USA and International Mathematical Olympiads 2003, edited byTitu Andreescu
andZumingFeng
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1622 fax:[phone]
www.maa.org

Contents
Preface ix
Acknowledgments xi
Abbreviations and Notations xiii
Introduction xv
1 The Problems 1
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Team Selection Test. . . . . . . . . . . . . . . . . . . . . 3
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Hints 7
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Team Selection Test. . . . . . . . . . . . . . . . . . . . . 8
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Formal Solutions 11
1 USAMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Team Selection Test. . . . . . . . . . . . . . . . . . . . . 32
3 IMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Problem Credits 61
5 Glossary 63
6 Further Reading 69
vii
viii USAandInternationalMathematicalOlympiads2003
7 Appendix 75
1 2002 Olympiad Results . . . . . . . . . . . . . . . . . . . 75
2 2002 Olympiad Results . . . . . . . . . . . . . . . . . . . 77
3 2001 Olympiad Results . . . . . . . . . . . . . . . . . . . 79
4 2000 Olympiad Results . . . . . . . . . . . . . . . . . . . 80
5 1999 Olympiad Results . . . . . . . . . . . . . . . . . . . 82
6 1999–2003 Cumulative IMO Results . . . . . . . . . . . 83
About the Authors 85
Preface
This book is intended tohelp students preparingtoparticipatein theUSA
Mathematical Olympiad (USAMO) in thehopeof representingtheUnited
States at the International Mathematical Olympiad (IMO). The USAMO
is the third stage of the selection process leading to participation in the
IMO.TheprecedingexaminationsaretheAMC10ortheAMC12(which
replaced the American High School Mathematics Examination) and the
American Invitational Mathematics Examination (AIME). Participation in
theAIMEandtheUSAMOisbyinvitationonly,basedonperformancein
the preceding exams of the sequence.
The top 12 USAMO students are invited to attend the Mathematical
Olympiad Summer Program (MOSP) regardless of their grade in school.
Additional MOSP invitations are extended to the most promising non-
graduatingUSAMOstudents,aspotentialIMOparticipantsinfutureyears.
During the first days of MOSP, IMO-type exams are given to the top 12
USAMOstudentswiththegoalofidentifyingthesixmembersoftheUSA
IMO Team. The Team Selection Test (TST) simulates an actual IMO,
consisting of six problems to be solved over two 4 1/2 hour sessions. The
12 equally weighted problems (six on the USAMO and six on the TST)
determine the USA Team.
The Mathematical Olympiad booklets have been published since 1976.
Copies for each year through 1999 can be ordered from the Mathematical
Association of American (MAA) American Mathematics Competitions
(AMC). This publication, Mathematical Olympiads 2000, Mathematical
Olympiads 2001, and Mathematical Olympiads 2002 are published bythe
MAA.Inaddition,variousotherpublicationsareusefulinpreparingforthe
AMC-AIME-USAMO-IMO sequence (see Chapter 6, Further Reading).
ix
x USAandInternationalMathematicalOlympiads2003
For more information about the AMC examinations, or to order Mathe-
matical Olympiad booklets from previous years, please write to
Steven Dunbar, MAA Director for K–12 Programs
American Mathematics Competitions
University of Nebraska-Lincoln
1740 Vine Street
Lincoln, NE 68588-0658,
or visit the AMC web site at www.unl.edu/amc.