Lecture Notes on Mathematical Olympiad Courses


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1 Lecture Notes on Mathematical Olympiad Courses For Junior Section Vol. 2
2 Mathematical Olympiad Series ISSN: Series Editors: Lee Peng Yee (Nanyang Technological University, Singapore) Xiong Bin (East China Normal University, China) Published Vol. 1 Vol. 2 A First Step to Mathematical Olympiad Problems by Derek Holton (University of Otago, New Zealand) Problems of Number Theory in Mathematical Competitions by Yu HongBing (Suzhou University, China) translated by Lin Lei (East China Normal University, China)
3 Xu Jiagu Vol. 6 Mathematical Olympiad Series Lecture Notes on Mathematical Olympiad Courses For Junior Section Vol. 2 World Scientific
4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library CataloguinginPublication 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 (pbk) (Set) ISBN (pbk) (Set) ISBN (pbk) (Vol. 1) ISBN (pbk) (Vol. 1) ISBN (pbk) (Vol. 2) ISBN (pbk) (Vol. 2) Printed in Singapore.
5 Preface Although mathematical olympiad competitions are carried out by solving problems, the system of Mathematical Olympiads and the related training courses cannot involve only the techniques of solving mathematical problems. Strictly speaking, it is a system of mathematical advancing education. To guide students who are interested in mathematics and have the potential to enter the world of Olympiad mathematics, so that their mathematical ability can be promoted efficiently and comprehensively, it is important to improve their mathematical thinking and technical ability in solving mathematical problems. An excellent student should be able to think flexibly and rigorously. Here the ability to do formal logic reasoning is an important basic component. However, it is not the main one. Mathematical thinking also includes other key aspects, like starting from intuition and entering the essence of the subject, through prediction, induction, imagination, construction, design and their creative abilities. Moreover, the ability to convert concrete to the abstract and vice versa is necessary. Technical ability in solving mathematical problems does not only involve producing accurate and skilled computations and proofs, the standard methods available, but also the more unconventional, creative techniques. It is clear that the usual syllabus in mathematical educations cannot satisfy the above requirements, hence the mathematical olympiad training books must be selfcontained basically. The book is based on the lecture notes used by the editor in the last 15 years for Olympiad training courses in several schools in Singapore, like Victoria Junior College, Hwa Chong Institution, Nanyang Girls High School and Dunman High School. Its scope and depth significantly exceeds that of the usual syllabus, and introduces many concepts and methods of modern mathematics. The core of each lecture are the concepts, theories and methods of solving mathematical problems. Examples are then used to explain and enrich the lectures, and indicate their applications. And from that, a number of questions are included for the reader to try. Detailed solutions are provided in the book. The examples given are not very complicated so that the readers can understand them more easily. However, the practice questions include many from actual v
6 vi Preface competitions which students can use to test themselves. These are taken from a range of countries, e.g. China, Russia, the USA and Singapore. In particular, there are many questions from China for those who wish to better understand mathematical Olympiads there. The questions are divided into two parts. Those in Part A are for students to practise, while those in Part B test students ability to apply their knowledge in solving real competition questions. Each volume can be used for training courses of several weeks with a few hours per week. The test questions are not considered part of the lectures, since students can complete them on their own. K. K. Phua
7 Acknowledgments My thanks to Professor Lee Peng Yee for suggesting the publication of this the book and to Professor Phua Kok Khoo for his strong support. I would also like to thank my friends, Ang Lai Chiang, Rong Yifei and Gwee Hwee Ngee, lecturers at HwaChong, Tan Chik Leng at NYGH, and Zhang Ji, the editor at WSPC for her careful reading of my manuscript, and their helpful suggestions. This book would be not published today without their efficient assistance. vii
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9 Abbreviations and Notations Abbreviations AHSME AIME APMO ASUMO BMO CHNMOL CHNMOL(P) CHINA CMO HUNGARY IMO IREMO KIEV MOSCOW POLAND PUTNAM RUSMO SSSMO SMO SSSMO(J) SWE American High School Mathematics Examination American Invitational Mathematics Examination Asia Pacific Mathematics Olympiad Olympics Mathematical Competitions of All the Soviet Union British Mathematical Olympiad China Mathematical Competition for Secondary Schools China Mathematical Competition for Primary Schools China Mathematical Competitions for Secondary Schools except for CHNMOL Canada Mathematical Olympiad Hungary Mathematical Competition International Mathematical Olympiad Ireland Mathematical Olympiad Kiev Mathematical Olympiad Moscow Mathematical Olympiad Poland Mathematical Olympiad Putnam Mathematical Competition AllRussia Olympics Mathematical Competitions Singapore Secondary Schools Mathematical Olympiads Singapore Mathematical Olympiads Singapore Secondary Schools Mathematical Olympiads for Junior Section Sweden Mathematical Olympiads ix
10 x Abbreviations and Notations USAMO USSR United States of American Mathematical Olympiad Union of Soviet Socialist Republics Notations for Numbers, Sets and Logic Relations N the set of positive integers (natural numbers) N 0 the set of nonnegative integers Z the set of integers Z + the set of positive integers Q the set of rational numbers Q + the set of positive rational numbers Q + 0 the set of nonnegative rational numbers R the set of real numbers [m, n] the lowest common multiple of the integers m and n (m, n) the greatest common devisor of the integers m and n a b a divides b x absolute value of x x the greatest integer not greater than x x the least integer not less than x {x} the decimal part of x, i.e. {x} = x x a b (mod c) a is congruent to b modulo c ( n k) the binomial coefficient n choose k n! n factorial, equal to the product n [a, b] the closed interval, i.e. all x such that a x b (a, b) the open interval, i.e. all x such that a < x < b iff, if and only if implies A B A is a subset of B A B the set formed by all the elements in A but not in B A B the union of the sets A and B A B the intersection of the sets A and B a A the element a belongs to the set A
11 Contents Preface Acknowledgments Abbreviations and Notations 16 Quadratic Surd Expressions and Their Operations 1 17 Compound Quadratic Surd Form a ± b 7 18 Congruence of Integers Decimal Representation of Integers Perfect Square Numbers Pigeonhole Principle x and {x} Diophantine Equations (I) Roots and Discriminant of Quadratic Equation ax 2 + bx + c = Relation between Roots and Coefficients of Quadratic Equations 61 v vii ix 26 Diophantine Equations (II) Linear Inequality and System of Linear Inequalities Quadratic Inequalities and Fractional Inequalities 83 xi
12 xii Contents 29 Inequalities with Absolute Values Geometric Inequalities 95 Solutions to Testing Questions 103 Index 177
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