1、Le Hai Chau Le Hai Khoi Mathematical Olympiad Series Vol.5 Selected Problems of the Vietnamese Mathematical Olympiad (1962-2009) World Scientific Selected Problems of the Vietnamese Mathematical Olympiad (19622009) 7514 tp.indd 1 8/3/10 9:49 AMVol. 5 Mathematical Olympiad Series Selected Problems of
2、 the Vietnamese Mathematical Olympiad (19622009) World Scientific Le Hai Chau Ministry of Education and Training, Vietnam Le Hai Khoi Nanyang Technological University, Singapore 7514 tp.indd 2 8/3/10 9:49 AMBritish Library Cataloguing-in-Publication Data A catalogue record for this book is available
3、 from the British Library. 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-13 978-981-4289-59-7 (pbk) ISBN-10
4、981-4289-59-0 (pbk) 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 Publis
5、her. Copyright 2010 by World Scientific Publishing Co. Pte. Ltd. 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 Printed in Singapor
6、e. Mathematical Olympiad Series Vol. 5 SELECTED PROBLEMS OF THE VIETNAMESE OLYMPIAD (19622009) LaiFun - Selected Problems of the Vietnamese.pmd 8/23/2010, 3:16 PM 1Mathematical Olympiad Series ISSN: 1793-8570 Series Editors: Lee Peng Yee (Nanyang Technological University, Singapore) Xiong Bin (East
7、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 Univers
8、ity, China) Vol. 3 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) Vol. 5 Selected Problems of the Vietnamese Olympiad (1962
9、2009) by Le Hai Chau (Ministry of Education and Training, Vietnam) & Le Hai Khoi (Nanyang Technology University, Singapore) Vol. 6 Lecture Notes on Mathematical Olympiad Courses: For Junior Section (In 2 Volumes) by Jiagu Xu LaiFun - Selected Problems of the Vietnamese.pmd 8/23/2010, 3:16 PM 2Forewo
10、rd The International Mathematical Olympiad (IMO) - an annual international mathematical competition primarily for high school students - has a his- tory of more than half a century and is the oldest of all international science Olympiads. Having attracted the participation of more than 100 countries
11、 and territories, not only has the IMO been instrumental in promoting inter- est in mathematics among high school students, it has also been successful in the identication of mathematical talent. For example, since 1990, at least one of the Fields Medalists in every batch had participated in an IMO
12、earlier and won a medal. Vietnam began participating in the IMO in 1974 and has consistently done very well. Up to 2009, the Vietnamese team had already won 44 gold, 82 silver and 57 bronze medals at the IMO - an impressive performance that places it among the top ten countries in the cumulative med
13、al tally. This is probably related to the fact that there is a well-established tradition in mathematical competitions in Vietnam - the Vietnamese Mathematical Olympiad (VMO) started in 1962. The VMO and the Vietnamese IMO teams have also helped to identify many outstanding mathematical talents from
14、 Vietnam, including Ngo Bao Chau, whose proof of the Fundamental Lemma in Langlands program made it to the list of Top Ten Scientic Discoveries of 2009 of Time magazine. It is therefore good news that selected problems from the VMO are now made more readily available through this book. One of the au
15、thors - Le Hai Chau - is a highly respected mathemat- ics educator in Vietnam with extensive experience in the development of mathematical talent. He started working in the Ministry of Education of Vietnam in 1955, and has been involved in the VMO and IMO as a set- ter of problems and the leader of
16、the Vietnamese team to several IMO. He has published many mathematics books, including textbooks for sec- ondary and high school students, and has played an important role in the development of mathematical education in Vietnam. For his contributions, he was bestowed the nations highest honour of “P
17、eoples Teacher” by the government of Vietnam in 2008. Personally, I have witnessed rst-hand the kind of great respect expressed by teachers and mathematicians in Vietnam whenever the name “Le Hai Chau” is mentioned. Le Hai Chaus passion for mathematics is no doubt one of the main reasons that his so
18、n Le Hai Khoi - the other author of this book - also fell vvi FOREWORD in love with mathematics. He has been a member of a Vietnamese IMO team, and chose to be a mathematician for his career. With a PhD in mathematics from Russia, Le Hai Khoi has worked in both Vietnam and Singapore, where he is bas
19、ed currently. Like his father, Le Hai Khoi also has a keen interest in discovering and nurturing mathematical talent. I congratulate the authors for the successful completion of this book. I trust that many young minds will nd it interesting, stimulating and enriching. San Ling Singapore, Feb 2010Pr
20、eface In 1962, the rst Vietnamese Mathematical Olympiad (VMO) was held in Hanoi. Since then the Vietnam Ministry of Education has, jointly with the Vietnamese Mathematical Society (VMS), organized annually (except in 1973) this competition. The best winners of VMO then participated in the Selection
21、Test to form a team to represent Vietnam at the International Mathematical Olympiad (IMO), in which Vietnam took part for the rst time in 1974. After 33 participations (except in 1977 and 1981) Vietnamese students have won almost 200 medals, among them over 40 gold. This books contains about 230 sel
22、ected problems from more than 45 competitions. These problems are divided into ve sections following the classication of the IMO: Algebra, Analysis, Number Theory, Combina- torics, and Geometry. It should be noted that the problems presented in this book are of average level of diculty. In the futur
23、e we hope to prepare another book containing more dicult problems of the VMO, as well as some problems of the Selection Tests for forming the Vietnamese teams for the IMO. We also note that from 1990 the VMO has been divided into two eche- lons. The rst echelon is for students of the big cities and
24、provinces, while the second echelon is for students of the smaller cities and highland regions. Problems for the second echelon are denoted with the letter B. We would like to thank the World Scientic Publishing Co. for publish- ing this book. Special thanks go to Prof. Lee Soo Ying, former Dean of
25、the College of Science, Prof. Ling San, Chair of the School of Physical and Mathematical Sciences, and Prof. Chee Yeow Meng, Head of the Division of Mathematical Sciences, Nanyang Technological University, Singapore, for stimulating encouragement during the preparation of this book. We are grateful
26、to David Adams, Chan Song Heng, Chua Chek Beng, Anders Gus- tavsson, Andrew Kricker, Sinai Robins and Zhao Liangyi from the School of Physical Mathematical Sciences, and students Lor Choon Yee and Ong Soon Sheng, for reading dierent parts of the book and for their valuable suggestions and comments t
27、hat led to the improvement of the exposition. We are also grateful to Lu Xiao for his help with the drawing of gures, and to Adelyn Le for her help in editing of some paragraphs of the book. We would like to express our gratitude to the Editor of the Series “Math- ematical Olympiad”, Prof. Lee Peng
28、Yee, for his attention to this work. We thank Ms. Kwong Lai Fun of World Scientic Publishing Co. for her viiviii PREFACE hard work to prepare this book for publication. We also thank Mr. Wong Fook Sung, Albert of Temasek Polytechnic, for his copyediting of the book. Last but not least, we are respon
29、sible for any typos, errors,. in the book, and hope to receive the readers feedback. The Authors Hanoi and Singapore, Dec 2009Contents Foreword v Preface vii 1 The Gifted Students 1 1.1 The Vietnamese Mathematical Olympiad . 1 1.2 High Schools for the Gifted in Maths . 1 0 1.3 Participating in IMO 1
30、 3 2B a s i cN o t i o n sa n dF a c t s 1 7 2.1 Algebra. 1 7 2.1.1 Important inequalities . 1 7 2.1.2 Polynomials 1 9 2.2 Analysis 2 0 2.2.1 Convex and concave functions 2 0 2.2.2 Weierstrass theorem 2 0 2.2.3 Functional equations 2 1 2.3 Number Theory . 2 1 2.3.1 Prime Numbers . 2 1 2.3.2 Modulo o
31、peration 2 3 2.3.3 Fermat and Euler theorems 2 3 2.3.4 Numeral systems 2 4 2.4 Combinatorics 2 4 2.4.1 Counting 2 4 2.4.2 Newton binomial formula . 2 5 2.4.3 Dirichlet (or Pigeonhole) principle . 2 5 2.4.4 Graph . 2 6 2.5 Geometry . 2 7 2.5.1 Trigonometric relationship in a triangle and a circle 2 7
32、 ixx CONTENTS 2.5.2 Trigonometric formulas 2 8 2.5.3 Some important theorems. 2 9 2.5.4 Dihedral and trihedral angles 3 0 2.5.5 Tetrahedra 3 1 2.5.6 Prism, parallelepiped, pyramid . 3 1 2.5.7 Cones 3 1 3P r o b l e m s 3 3 3.1 Algebra. 3 3 3.1.1 (1962) . . . . . . . . . . . . . . . . . . . . . . . .
33、 . . 33 3.1.2 (1964) . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.3 (1966) . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.4 (1968) . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.5 (1969) . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.6 (1970) . . . .
34、 . . . . . . . . . . . . . . . . . . . . . . 34 3.1.7 (1972) . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.8 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.9 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.10 (1976) . . . . . . . . . . . . . . . . .
35、. . . . . . . . . 35 3.1.11 (1976) . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.12 (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.13 (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.14 (1978) . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1
36、.15 (1978) . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.16 (1979) . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.17 (1979) . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.18 (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.19 (1980) . . . . . . .
37、 . . . . . . . . . . . . . . . . . . . 37 3.1.20 (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.21 (1981) . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.22 (1981) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.23 (1981) . . . . . . . . . . . . . . . . . . .
38、 . . . . . . . 38 3.1.24 (1981) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.25 (1982) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.26 (1982) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.27 (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.28
39、 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.29 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.30 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.31 (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.32 (1986) . . . . . . . .
40、. . . . . . . . . . . . . . . . . . 40 3.1.33 (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . 40CONTENTS xi 3.1.34 (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.35 (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.36 (1989) . . . . . . . . . . . . . . .
41、 . . . . . . . . . . . 41 3.1.37 (1990 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.38 (1991 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.39 (1992 B) . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.40 (1992 B) . . . . . . . . . . . . . . . . . . . . . . . . . 4
42、1 3.1.41 (1992) . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.42 (1994 B) . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.43 (1994) . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.44 (1995) . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.45 (1996) . . . .
43、. . . . . . . . . . . . . . . . . . . . . . 42 3.1.46 (1996) . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.47 (1997) . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.48 (1998 B) . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.49 (1998) . . . . . . . . . . . . . . . .
44、. . . . . . . . . . 43 3.1.50 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.51 (1999) . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.52 (2001 B) . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.53 (2002) . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3
45、.1.54 (2003) . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.55 (2004 B) . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.56 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.57 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.58 (2005) . . . . . .
46、 . . . . . . . . . . . . . . . . . . . . 45 3.1.59 (2006 B) . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.60 (2006 B) . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.61 (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.62 (2007) . . . . . . . . . . . . . . . . . .
47、 . . . . . . . . 46 3.1.63 (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Analysis 4 7 3.2.1 (1965) . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.3 (1980) . . . . . . . . . . . . . . . . . . . . . . .
48、 . . . 47 3.2.4 (1983) . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.5 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.6 (1984) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.7 (1985) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.8 (1986) . . .
49、 . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.9 (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.10 (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.11 (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . 49xii CONTENTS 3.2.12 (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.13 (1989) . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.14 (1990 B) . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.15 (1990) . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.16 (1990) . . . . . . .
