Lecture Notes on Mathematical Olympiad Courses

For Junior Section Vol. 2

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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.pmd 11/2/2009, 3:35 PM2

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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)

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Mathematical Olympiad Series — Vol. 6 LECTURE NOTES ON MATHEMATICAL OLYMPIAD COURSES For Junior Section

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Preface

Although mathematical olympiad competitions are carried out by solving prob- lems, the system of Mathematical Olympiads and the related training courses can- not involve only the techniques of solving mathematical problems. Strictly speak- ing, 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 tech- nical 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 pro- ducing accurate and skilled computations and proofs, the standard methods avail- able, 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 self-contained 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 under- stand them more easily. However, the practice questions include many from actual

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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 mathe- matical 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

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.

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Abbreviations and Notations

Abbreviations

AHSME American High School Mathematics Examination AIME American Invitational Mathematics Examination

APMO Asia Pacific Mathematics Olympiad ASUMO Olympics Mathematical Competitions of All

the Soviet Union BMO British Mathematical Olympiad

CHNMOL China Mathematical Competition for Secondary Schools

CHNMOL(P) China Mathematical Competition for Primary Schools

CHINA China Mathematical Competitions for Secondary Schools except for CHNMOL

CMO Canada Mathematical Olympiad HUNGARY Hungary Mathematical Competition

IMO International Mathematical Olympiad IREMO Ireland Mathematical Olympiad

KIEV Kiev Mathematical Olympiad MOSCOW Moscow Mathematical Olympiad POLAND Poland Mathematical Olympiad PUTNAM Putnam Mathematical Competition

RUSMO All-Russia Olympics Mathematical Competitions SSSMO Singapore Secondary Schools Mathematical Olympiads

SMO Singapore Mathematical Olympiads SSSMO(J) Singapore Secondary Schools Mathematical Olympiads

for Junior Section SWE Sweden Mathematical Olympiads

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x Abbreviations and Notations

USAMO United States of American Mathematical Olympiad USSR Union of Soviet Socialist Republics

Notations for Numbers, Sets and Logic Relations

N the set of positive integers (natural numbers) N0 the set of non-negative 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 non-negative 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 bxc the greatest integer not greater than x dxe the least integer not less than x {x} the decimal part of x, i.e. {x} = x− bxc

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 1 · 2 · 3 · 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

Contents

Preface v

Acknowledgments vii

Abbreviations and Notations ix

16 Quadratic Surd Expressions and Their Operations 1

17 Compound Quadratic Surd Form √

a ± √b 7

18 Congruence of Integers 13

19 Decimal Representation of Integers 19

20 Perfect Square Numbers 25

21 Pigeonhole Principle 31

22 bxc and {x} 37

23 Diophantine Equations (I) 45

24 Roots and Discriminant of Quadratic Equation ax2 + bx + c = 0 53

25 Relation between Roots and Coefficients of Quadratic Equations 61

26 Diophantine Equations (II) 69

27 Linear Inequality and System of Linear Inequalities 77

28 Quadratic Inequalities and Fractional Inequalities 83

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xii Contents

29 Inequalities with Absolute Values 89

30 Geometric Inequalities 95

Solutions to Testing Questions 103

Index 177

Lecture 16

Quadratic Surd Expressions and Their Operations

Definitions

For an even positive integer n, by the notation n √

a, where a ≥ 0, we denote the non-negative real number x which satisfies the equation xn = a. In particular, when n = 2, 2

√ a is called square root of a, and denoted by

√ a usually.

For odd positive integer n and any real number a, by the notation n √

a we denote the r