Inequalities: An Approach Through Problems

The International Mathematical Olympiad(IMO), which started as a simple contest among seven communist block countries in Europe in 1959, has now encompassed the whole world. With nearly 100 countries participating in this mega event, this has acquired a true international character. Mathematical olympiad has effused new enthusiasm in the last few generations of young stu- dents and really talented young minds have started getting attracted to the rare beauty of mathematics. Along with it, new ideas have emerged and many intricate problems woven around these ideas have naturally been discovered. This has enriched basic mathematics, strengthened the foundations of elementary mathematics and has posed challenging problems to the younger generation. In turn, high school mathematics has undergone a profound change. Even though the concept of inequalities is old, the mathematical olympiad movement has generated new problems based on these inequalities and newer applications of these old inequalities. There are classical inequalities like the Arithmetic mean-Geometric mean inequality and the Cauchy-Schwarz inequal- ity wh ich are very old and which have innumerable applications. The main purpose of this book is to give a comprehensive presentation of inequalities and their use in mathematical olympiad problems. This book is also intended for those students who would like to participate in mathematical olympiad. Hopefully, this will fill a little vacuum that exists in the world of books. I have not touched on integral inequalities; they are not apart of olympiad mathematics.