Olympiad mathematics is not a collection of techniques of solving mathematical problems but a system for advancing mathematical education. This book is based on the lecture notes of the mathematical Olympiad training courses conducted by the author in Singapore. Its scope and depth not only covers and beyond the usual syllabus, but introduces a variety of concepts and methods in modern mathematics as well.In each lecture, the concepts, theories and methods are taken as the core. The examples serve to explain and enrich their intentions and to indicate their applications. Besides, appropriate number of test questions is available for the readers' practice and testing purpose. Their detailed solutions are also conveniently provided.The examples are not very complicated so readers can easily understand. There are many real competition questions included which students can use to verify their abilities. These test questions originate from many countries all over the world. This book will serve as a useful textbook of mathematical Olympiad courses, a self-study lecture notes for students, or as a reference book for related teachers and researchers.
Volume 1: Fractional Equations; Higher Degree Polynomial Equations; Irrational Equations; Indicial Functions and Logarithmic Functions; Trigonometric Functions; Law of Sines and Law of Cosines; Manipulations of Trigonometric Expressions; Extreme Values of Functions and Mean Inequality; Extreme Value Problems in Trigonometry; Fundamental Properties of Circles; Relation of Line and Circle and Relation of Circles; Cyclic Polygons; Power of a Point with Respect to a Circle; Some Important Theorems in Geometry; Five Centers of a Triangle; Volume 2: Mathematical Induction; Arithmetic Progression and Geometric Progression; Recursive Sequence; Summation of Series; Some Fundamental Theorems on Congruence; Chinese Remainder Theorem and Order of Integer; Diophantine Equation (III); Cauchy - Schwartz Inequality; Rearrangement Inequality and Jensen' Inequality; Schur Inequality; Fractional Inequality; Variable - Freezing Method; Some Methods in Counting Numbers (I); Some Methods in Counting Numbers (II); Introduction to Functional Equations.