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Lecture 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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 problems,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