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Chapter 1 Mathematical Olympiad in China 1

1.1 International Mathematical Olympiad(IMO)and China Mathematical Contest—Written before the 31st IMO 1

1.1.1 A Brief Introduction to IMO 2

1.1.2 A Historic Review of China Mathematical Contest 4

1.1.3 Activities of China in the IMO and the 31st IMO 6

Chapter 2 Olympiad's Mathematics 8

2.1 The Application of Projective Geometry Methods to Problem Proving in Geometry 8

2.1.1 A Few Concepts in Projective Geometry 10

2.1.2 Some Examples 15

2.1.3 Exercises 22

2.2 A Conjecture Concerning Six Points in a Square 24

2.3 Modulo-Period Sequence of Numbers 33

2.3.1 Basic Concepts 33

2.3.2 Pure Modulo-period Sequence 39

2.3.3 The Periodicity of Sum Sequence 44

2.3.4 The Relation between the Period and the Initial Terms 47

2.4 Iteration of Fractional Linear Function and Consturction of a Class of Function Equation 49

2.5 Remarks Initiating from a Putnam Mathematics Competition Problem 55

2.5.1 Introductory Remarks 55

2.5.2 The Proof of the Problem 56

2.5.3 Reinforcing the Promble 57

2.5.4 Application 60

2.5.5 Mutually Supplementary Sequences and Reversible Sequences 64

2.6 The Ways of Finding the Best Choise Point 67

2.6.1 The Congruent Transformation of Figures 67

2.6.2 Similarity Transformation of Figures 69

2.6.3 Partial Adjusting Method 70

2.6.4 The Contour Line Method 73

2.6.5 Algebraic Method 75

2.6.6 Trigonometrical Method 77

2.6.7 Analytic Method 78

2.6.8 Solution by Fermat Point Theorem 79

2.6.9 The Area Method 80

2.6.10 Physical Method 81

2.7 The Formulas and Inequalities for the Volumes of n-Simplex 84

2.8 The Polynomial of Inverse Root and Its Transformation 100

2.8.1 The Extension of an IMO Problem 100

2.8.2 The Inverse Root Polynomial 102

2.8.3 Trigonometric Formula of Recurrence Type 105

2.8.4 Inverse Root Polynomial Transformation 108

Chapter 3 Suggestions and Answers of Problems 116

3.1 Remarks on Proposing Problems for Mathematics Competition 116

3.2 A Problem of IMO and a Useful Polynomial 131

3.2.1 Introduction 131

3.2.2 The Proof of the Problem 132

3.2.3 Some Properties of Fm(x) 135

3.2.4 Fm(x)and Some IMO Problems 138

3.2.5 An Existence Problem 142

3.3 Preliminary Approach to Methods of Proposing Mathematics Competition Problems 144

3.3.1 Introduction 145

3.3.2 Form Changing 148

3.3.3 Generalization 151

3.3.4 Construction 156

Chapter 4 Comment on the Exam Paper of Mathematical Olympiad Winter Camp in China 159

4.1 Comment on the Exam Paper of the First Mathematical Winter Camp(1986) 159

4.2 Comment on the Exam Paper of the Second Mathematical Winter Camp(1987) 172

4.3 Comment on the Exam Paper of the Third Mathematical Winter Camp(1988) 178

4.4 Comment on the Exam Paper of the Fourth Mathematical Winter Camp(1989) 183

4.5 Comment on the Exam Paper of the Fifth Mathematical Winter Camp(1990) 192

Chapter 5 China Mathematical Olympiad from the First to the Lastest 205

5.1 China Mathematical Olympiad(1991) 205

5.2 China Mathematical Olympiad(1992) 212

5.3 China Mathematical Olympiad(1993) 220

5.4 China Mathematical Olympiad(1994) 225

5.5 China Mathematical Olympiad(1995) 235

5.6 China Mathematical Olympiad(1996) 241

5.7 China Mathematical Olympiad(1997) 247

5.8 China Mathematical Olympiad(1998) 258

5.9 China Mathematical Olympiad(1999) 265

5.10 China Mathematical Olympiad(2000) 275

5.11 China Mathematical Olympiad(2001) 282

5.12 China Mathematical Olympiad(2002) 293

5.13 China Mathematical Olympiad(2003) 304

5.14 China Mathematical Olympiad(2004) 316

5.15 China Mathematical Olympiad(2005) 323

5.16 China Mathematical Olympiad(2006) 333

5.17 China Mathematical Olympiad(2007) 342

5.18 China Mathematical Olympiad(2008) 351

5.19 China Mathematical Olympiad(2009) 360

5.20 China Mathematical Olympiad(2010) 369

5.21 China Mathematical Olympiad(2011) 375

5.22 China Mathematical Olympiad(2012) 381

5.23 China Mathematical Olympiad(2013) 388