Lecture Notes on Mathematical Olympiad Courses

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1 Lecture Notes on Mathematical Olympiad Courses For Senior Section Vol. 1

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 Vol. 3 Vol. 4 A First Step to Mathematical Olympiad Problems by Derek Holton (University of Otago, New Zealand) Problems of Number Theory in Mathematical Competitions by Yu Hong-Bing (Suzhou University, China) translated by Lin Lei (East China Normal University, China) Graph Theory by Xiong Bin (East China Normal University, China) & Zheng Zhongyi (High School Attached to Fudan University, China) translated by Liu Ruifang, Zhai Mingqing & Lin Yuanqing (East China Normal University, China) Combinatorial Problems in Mathematical Competitions by Yao Zhang (Hunan Normal University, P. R. China) Vol. 5 Selected Problems of the Vietnamese Olympiad ( ) by Le Hai Chau (Ministry of Education and Training, Vietnam) & Le Hai Khoi (Nanyang Technology University, Singapore) Vol. 6 Vol. 7 Lecture Notes on Mathematical Olympiad Courses: For Junior Section (In 2 Volumes) by Xu Jiagu A Second Step to Mathematical Olympiad Problems by Derek Holton (University of Otago, New Zealand & University of Melbourne, Australia) Vol. 8 Lecture Notes on Mathematical Olympiad Courses: For Senior Section (In 2 Volumes) by Xu Jiagu

3 Xu Jiagu Former Professor of Mathematics, Fudan University, China Vol. 8 Mathematical Olympiad Series Lecture Notes on Mathematical Olympiad Courses For Senior Section Vol. 1 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 Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Mathematical Olympiad Series Vol. 8 LECTURE NOTES ON MATHEMATICAL OLYMPIAD COURSES For Senior Section (In 2 Volumes) Copyright 2012 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 consist only of problem solving techniques. Strictly speaking, it is a system of mathematical advancing education. To guide students, who are interested in 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 perform formal logic reasoning is an important basic component. However, it is not the main one. Mathematical thinking also includes other key aspects, such as starting from intuition and entering the essence of the subject, through the processes of prediction, induction, imagination, construction and design to conduct their creative activities. In addition, the ability to convert the concrete to the abstract and vice versa is essential. Technical ability in solving mathematical problems does not only involve producing accurate and skilled-computations and proofs using the standard methods available, but also the more unconventional, creative techniques. It is clear that the standard syllabus in mathematical education cannot satisfy the above requirements. Hence the Mathematical Olympiad training books must be self-contained basically. This 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, such as Victoria Junior College, Hwa Chong Institution, Nanyang Girls High School and Dunman High School. Its scope and depth significantly exceeds that of the standard syllabus provided in schools, and introduces many concepts and methods from modern mathematics. v

6 vi Preface 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, as well as to indicate the applications of these concepts and methods. A number of questions are included at the end of each lecture for the reader to try. Detailed solutions are provided at the end of book. The examples given are not very complicated so that the readers can understand them easily. However, many of the practice questions at the end of lectures are taken from actual competitions, which students can use to test themselves. These questions are taken from a range of countries, such as China, Russia, the United States of America and Singapore. In particular, there are many questions from China for those who wish to better understand Mathematical Olympiads there. The questions at the end of each lecture 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 as students can complete them on their own.

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 friend Mr Fu Ling Chen, lecturer at TJC for his corrections, as well as Zhang Ji and He Yue, the editors of this book at World Scientific Publishing Co. (WSPC). This book would not have been published without their efficient assistance. vii

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9 Abbreviations and Notations Abbreviations AHSME AIME APMO ASUMO AUSTRALIA AUSTRIA BALKAN BALTIC WAY BELARUS BMO BULGARIA CGMO CHINA CHNMO CHNMOL CMC CMO CNMO COLUMBIA CROATIA CSMO CWMO CZECH-POLISH-SLOVAK ESTONIA FINLAND GERMANY American High School Mathematics Examination American Invitational Mathematics Examination Asia Pacific Mathematics Olympiad Olympics Mathematical Competitions of All the Soviet Union Australia Mathematical Competitions Austria Mathematical Olympiad Balkan Mathematical Olympiad Baltic Way International Mathematical Competition Belarus Mathematical Olympiad British Mathematical Olympiad Bulgaria Mathematical Olympiad China Girl s Mathematical Olympiad China Mathematical Competitions for Secondary Schools except for CHNMOL China Mathematical Olympiad China Mathematical Competition for Secondary Schools China Mathematical Competition and its preliminary round Canada Mathematical Olympiad China Northern Mathematical Olympiad Columbia Mathematical Olympiad Croatia Mathematical Olympiad China Southeastern Mathematical Olympiad China Western Mathematical Olympiad International Competitions Czech-Polish-Slovak Match Estonia Mathematical Olympiad Finland Mathematical Olympiad Germany Mathematical Olympiad ix

10 x Abbreviations and Notations GREECE HONG KONG HUNGARY IMO INDIA IRAN IRE ITALY JAPAN KOREA KOREAN MC MACAO MOLDOVA NEW ZEALAND NORTH-EUROPEAN POLAND ROMANIA RUSMO SLOVENIA SSSMO SMO SSSMO(J) THAILAND TURKEY TST USAMO VIETNAM Greece Mathematical Olympiad Hong Kong Mathematical Olympiad Hungary Mathematical Competition International Mathematical Olympiad India Mathematical Olympiad Iran Mathematical Olympiad Ireland Mathematical Olympiad Italy Mathematical Olympiad Japan Mathematical Olympiad Korea Mathematical Olympiad Korean Mathematical Competition Macao Mathematical Olympiad Moldova Mathematical Olympiad New-Zealand Mathematical Olympiad North-European Mathematical Olympiad Poland Mathematical Olympiad Romania Mathematical Olympiad All-Russia Olympics Mathematical Competitions Slovenia Mathematical Olympiad Singapore Secondary Schools Mathematical Olympiads for Senior Section Singapore Mathematical Olympiads Singapore Secondary Schools Mathematical Olympiads for Junior Section Thailand Mathematical Olympiads Turkey Mathematical Olympiad Team Selection Test (including related training tests) United States of American Mathematical Olympiad Vietnam Mathematical Olympiad

11 Abbreviations and Notations xi Notations for Numbers, Sets and Logic Relations N the set of positive integers (natural numbers) the set of non-negative integers Z the set of integers the set of positive integers Q the set of rational numbers the set of positive rational numbers the set of non-negative rational numbers R the set of real numbers the set of positive real numbers N 0 Z C Q C Q C 0 R C R C 0 the set of non-negative real numbers Œa; 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 2 A the element a belongs to the set A

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13 Contents Preface v Acknowledgments vii Abbreviations and Notations ix 1 Fractional Equations 1 2 Higher Degree Polynomial Equations 9 3 Irrational Equations 17 4 Indicial Functions and Logarithmic Functions 25 5 Trigonometric Functions 31 6 Law of Sines and Law of Cosines 39 7 Manipulations of Trigonometric Expressions 47 8 Extreme Values of Functions and Mean Inequality 53 9 Extreme Value Problems in Trigonometry Fundamental Properties of Circles Relation of Line and Circle and Relation of Circles Cyclic Polygons Power of a Point with Respect to a Circle Some Important Theorems in Geometry 101 xiii

14 xiv Contents 15 Five Centers of a Triangle 109 Solutions to Testing Questions 117 Appendices 223 A Trigonometric Identities 225 B Mean Inequality 229 C Some Basic Inequalities Involving a Triangle 231 D Proofs of Some Important Theorems in Geometry 235 Index 245

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