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.
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.
目录:
Preface
Acknowledgments
Abbreviations and Notations
16 Mathematical Induction
17 Arithmetic Progressions and Geometric Progressions
18 Recursive Sequences
19 Summation of Various Sequences
20 Some Fundamental Theorems on Congruence
21 Chinese Remainder Theorem and Order of Integer
22 Diophantine Equations (Ⅲ)
23 Pythagorean Triples and Pell\'s Equations
24 Quadratic Residues
25 Some Important Inequalities(Ⅰ)
26 Somelmportantlnequalities(Ⅱ)
27 Some Methods For Solvinglnequalities
28 Some Basic Methods in Counting (Ⅰ)
29 Some Basic Methods in Counting (Ⅱ)
30 Introduction to Functional Equations
Solutions to Testing Questions
Appendices
A Theorem on Second Order Recursive Sequences
B Proofs of Tbeorems On Pell\'s Equation
C Theorems On Quadratic Residues
D Proofs of Some Important Inequalities
E Note On Cauchy\'s Problem in Functional Equations