Table Of ContentLecture Notes on
Mathematical
Olympiad Courses
For Junior Section Vol. 2
7600 tp.indd 3 11/4/09 1:57:55 PM
Mathematical Olympiad Series
ISSN: 1793-8570
Series Editors: Lee Peng Yee (Nanyang Technological University, Singapore)
Xiong Bin (East China Normal University, China)
Published
Vol. 1 A First Step to Mathematical Olympiad Problems
by Derek Holton (University of Otago, New Zealand)
Vol. 2 Problems of Number Theory in Mathematical Competitions
by Yu Hong-Bing (Suzhou University, China)
translated by Lin Lei (East China Normal University, China)
ZhangJi - Lec Notes on Math's Olymp Courses.p2md 11/2/2009, 3:35 PM
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
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Mathematical Olympiad Series — Vol. 6
LECTURE NOTES ON MATHEMATICAL OLYMPIAD COURSES
For Junior Section
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,
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ISBN-13 978-981-4293-53-2 (pbk) (Set)
ISBN-10 981-4293-53-9 (pbk) (Set)
ISBN-13 978-981-4293-54-9 (pbk) (Vol. 1)
ISBN-10 981-4293-54-7 (pbk) (Vol. 1)
ISBN-13 978-981-4293-55-6 (pbk) (Vol. 2)
ISBN-10 981-4293-55-5 (pbk) (Vol. 2)
Printed in Singapore.
ZhangJi - Lec Notes on Math's Olymp Courses.p1md 11/2/2009, 3:35 PM
Preface
Although mathematical olympiad competitions are carried out by solving prob-
lems,thesystemofMathematicalOlympiadsandtherelatedtrainingcoursescan-
notinvolveonlythetechniquesofsolvingmathematicalproblems. Strictlyspeak-
ing,itisasystemofmathematicaladvancingeducation.Toguidestudentswhoare
interested in mathematics and havethe potential to enter the worldof Olympiad
mathematics, so that their mathematical ability can be promoted efficiently and
comprehensively,itisimportanttoimprovetheirmathematicalthinkingandtech-
nicalabilityinsolvingmathematicalproblems.
Anexcellentstudentshouldbeabletothinkflexiblyandrigorously. Herethe
abilitytodoformallogicreasoningisanimportantbasiccomponent. However,it
isnotthemainone. Mathematicalthinkingalsoincludesotherkeyaspects, like
startingfromintuitionandenteringtheessenceofthesubject,throughprediction,
induction,imagination,construction,designandtheircreativeabilities.Moreover,
theabilitytoconvertconcretetotheabstractandviceversaisnecessary.
Technicalabilityinsolvingmathematicalproblemsdoesnotonlyinvolvepro-
ducingaccurateandskilledcomputationsandproofs,thestandardmethodsavail-
able,butalsothemoreunconventional,creativetechniques.
It is clear that the usual syllabus in mathematical educations cannot satisfy
theaboverequirements,hencethemathematicalolympiadtrainingbooksmustbe
self-containedbasically.
Thebookisbasedonthelecturenotesusedbytheeditorinthelast15yearsfor
Olympiad training courses in several schools in Singapore, like Victoria Junior
College, HwaChongInstitution, NanyangGirlsHighSchoolandDunmanHigh
School. Its scope and depth significantly exceeds that of the usual syllabus, and
introducesmanyconceptsandmethodsofmodernmathematics.
The core of each lecture are the concepts, theories and methods of solving
mathematicalproblems.Examplesarethenusedtoexplainandenrichthelectures,
andindicatetheirapplications. Andfromthat,anumberofquestionsareincluded
forthereadertotry. Detailedsolutionsareprovidedinthebook.
The examples given are not very complicated so that the readers can under-
standthemmoreeasily.However,thepracticequestionsincludemanyfromactual
v
vi Preface
competitions which students can use to test themselves. These are taken from a
rangeofcountries,e.g.China,Russia,theUSAandSingapore.Inparticular,there
are many questions from China for those who wish to better understand mathe-
maticalOlympiadsthere. Thequestionsaredividedintotwoparts. ThoseinPart
Aareforstudentstopractise,whilethoseinPartBteststudents’abilitytoapply
theirknowledgeinsolvingrealcompetitionquestions.
Each volume can be used for training courses of several weeks with a few
hoursperweek. Thetestquestionsarenotconsideredpartofthelectures, since
studentscancompletethemontheirown.
K.K.Phua
Acknowledgments
My thanks to Professor Lee Peng Yee for suggesting the publication of this the
bookandtoProfessorPhuaKokKhooforhisstrongsupport. Iwouldalsoliketo
thankmyfriends,AngLaiChiang,RongYifeiandGweeHweeNgee,lecturersat
HwaChong,TanChikLengatNYGH,andZhangJi,theeditoratWSPCforher
carefulreadingofmymanuscript,andtheirhelpfulsuggestions. Thisbookwould
benotpublishedtodaywithouttheirefficientassistance.
vii
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Abbreviations and Notations
Abbreviations
AHSME AmericanHighSchoolMathematicsExamination
AIME AmericanInvitationalMathematicsExamination
APMO AsiaPacificMathematicsOlympiad
ASUMO OlympicsMathematicalCompetitionsofAll
theSovietUnion
BMO BritishMathematicalOlympiad
CHNMOL ChinaMathematicalCompetitionforSecondary
Schools
CHNMOL(P) ChinaMathematicalCompetitionforPrimary
Schools
CHINA ChinaMathematicalCompetitionsforSecondary
SchoolsexceptforCHNMOL
CMO CanadaMathematicalOlympiad
HUNGARY HungaryMathematicalCompetition
IMO InternationalMathematicalOlympiad
IREMO IrelandMathematicalOlympiad
KIEV KievMathematicalOlympiad
MOSCOW MoscowMathematicalOlympiad
POLAND PolandMathematicalOlympiad
PUTNAM PutnamMathematicalCompetition
RUSMO All-RussiaOlympicsMathematicalCompetitions
SSSMO SingaporeSecondarySchoolsMathematicalOlympiads
SMO SingaporeMathematicalOlympiads
SSSMO(J) SingaporeSecondarySchoolsMathematicalOlympiads
forJuniorSection
SWE SwedenMathematicalOlympiads
ix
x AbbreviationsandNotations
USAMO UnitedStatesofAmericanMathematicalOlympiad
USSR UnionofSovietSocialistRepublics
NotationsforNumbers,SetsandLogicRelations
N thesetofpositiveintegers(naturalnumbers)
N thesetofnon-negativeintegers
0
Z thesetofintegers
Z+ thesetofpositiveintegers
Q thesetofrationalnumbers
Q+ thesetofpositiverationalnumbers
Q+ thesetofnon-negativerationalnumbers
0
R thesetofrealnumbers
[m,n] thelowestcommonmultipleoftheintegersmandn
(m,n) thegreatestcommondevisoroftheintegersmandn
a|b adividesb
|x| absolutevalueofx
(cid:98)x(cid:99) thegreatestintegernotgreaterthanx
(cid:100)x(cid:101) theleastintegernotlessthanx
{x} thedecimalpartofx,i.e. {x}=x−(cid:98)x(cid:99)
a≡b(modc) aiscongruenttobmoduloc
(cid:161) (cid:162)
n thebinomialcoefficientnchoosek
k
n! nfactorial,equaltotheproduct1·2·3·n
[a,b] theclosedinterval,i.e. allxsuchthata≤x≤b
(a,b) theopeninterval,i.e. allxsuchthata<x<b
⇔ iff,ifandonlyif
⇒ implies
A⊂B AisasubsetofB
A−B thesetformedbyalltheelementsinAbutnotinB
A∪B theunionofthesetsAandB
A∩B theintersectionofthesetsAandB
a∈A theelementabelongstothesetA
