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Published byEast China Normal University Press3663 North Zhongshan RoadShanghai 200062ChinaandWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

COMBINATORIAL PROBLEMS IN MATHEMATICAL COMPETITIONSCopyright 2011 by East China Normal University Press andWorld Scientific Publishing Co. Pte. Ltd.All rights reserved. This book, or paris thereof, may not be reproduced in any/orm or by any means,electronic or mechanical, including photocopying, recording or any in/ormation storage and retrievalsystem now known or to be invented, without writ/en permissionjromthe Publisher.

For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.

ISBN-13 978-981-283-949-7 (pbk)ISBN-IO 981-283-949-6 (pbk)

Printed in Singapore by Mainland Press Pte Ltd.

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Copy EditorsNI Ming East China Normal University Press, ChinaZHANG Ji World Scientific Publishing Co., SingaporeWONG Fook Sung Temasek Polytechnic, SingaporeKONG Lingzhi East China Normal University Press, China

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Introduction

This book consists of three parts: fundamental knowledge, basicmethods and typical problems. These three parts introduce thefundamental knowledge of solving combinatorial problems, theimportant solutions to combinatorial problems and some typicalproblems with often-used solutions in the high school mathematicalcompetition respectively.

In each chapter there are necessary examples and exercises withsolutions. These examples and exercises are of the same level ofdifficulty as the China Mathematical League Competitions which areselected from mathematical competitions at home and abroad in recentyears. Some test questions are created by the author himself and a feweasy questions in China Mathematical Olympiad (CMO) and IMO arealso included. In this book, the author pay attention to leading readersto explore, analyze and summarize the ideas and methods of solvingcombinatorial problems. The readers' mathematical concepts andabilities will be improved remarkably after acquiring knowledge fromthis book.

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Preface

Combinatorial Mathematics has a long history. Thousands ofyears ago, some simple and interesting combinatorial problems startedto be involved in the ancient Chinese masterpieces such as He Tu andLuo Shu. In the recent 20 years, owing to the rapid development ofknow ledge such as Computer Science, Coding Theory, ProgrammingTheory, Digital Communication and Experimental Designing, a seriesof theoretical and practical problems need to be solved by discretemathematics. Moreover, the questions which were raised by the logicof combinatorial mathematics itself and other branches ofmathematics, has made the research about combinatorial mathematicsto flourish and very rich. The problem-solving techniques and methodsare daedal, transforming this ancient mathematical idea into a rigorousmathematical subject.

The combinatorial questions in the mathematical competitions arealways straightforward, but the people who solve these problemsshould be endowed with the power of acute observation, richimagination and necessarily skills . There are no fixed methods that canbe followed, and various questions of different level of difficulties arevery rich. Hence, combinatorial questions have been included indifferent levels of intelligence test and mathematical competition.

This book focus on the combinatorial problems in the mathematicscompetition, and consists of three parts: fundamental knowledge,basic methods and typical problems. This book is a translation fromthe author 's book of the same title (in Chinese), with a few amendments

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x Combinatorial Problems in Mathematical Competitions

and supplements. I t not only emphasizes the fundamental knowledgeof solving combinatorial problems in mathematical competitions, butalso introduces to the readers the important solutions to combinatorialproblems and some typical problems with solutions that are often used.

In the choice of examples and exercises, except the questionswhich strikingly newer have been selected as far as possible, but thenovel test questions which are the same difficult level as the ChinaMathematical League Competitions in the mathematical competition athome and abroad in recent yeas have been chosen deliberately. Sometest questions created by the author himself and a few easy questions inChina Mathematical Olympiad (CMO) and IMO are also included. Inthis book, the author pays attention to leading the readers to explore,analyze and summarize the ideas and methods of solving combinatorialproblems. The readers' mathematical concepts and abilities will beimproved remarkably after acquiring the knowledge from this book.