From the Training of the USA IMO Team Titu Andreescu Zuming Feng
CONTENTS
1 Trigonometric Fundamentals 1 Definitions of Trigonometric Functions in Terms of Right Triangles 1 Think Within the Box 4 You’ve Got the Right Angle 6 Think Along the Unit Circle 10 Graphs of Trigonometric Functions 14 The Extended Law of Sines 18 Area and Ptolemy’s Theorem 19 Existence, Uniqueness, and Trigonometric Substitutions 23 Ceva’s Theorem 28 Think Outside the Box 33 Menelaus’s Theorem 33 The Law of Cosines 34 Stewart’s Theorem 35 Heron’s Formula and Brahmagupta’s Formula 37 Brocard Points 39 Vectors 41 The Dot Product and the Vector Form of the Law of Cosines 46 The Cauchy–Schwarz Inequality 47 Radians and an Important Limit 47 Constructing Sinusoidal Curves with a Straightedge 50 Three Dimensional Coordinate Systems 51 Traveling on Earth 55 Where Are You? 57 De Moivre’s Formula 58 2 Introductory Problems 63 3 Advanced Problems 73 4 Solutions to Introductory Problems 83 5 Solutions to Advanced Problems 125