# Olympiad inequalities book

**Posted:**July 10, 2007

**Filed under:**Notices 5 Comments

Author: Pham Van Thuan, Le Vi

This book is intended as a useful resource for high school and college students

who are training for national or international mathematical competitions.

But anybody who is interested in elementary mathematical inequalities may find

this book useful. This problem solving book is divided into six chapters,

containing more than fifty topics of interest to mathematical olympiad contestants

and coaches, demonstrating ideas and strategies in solving inequalities.

The reader will find in the book clever applications of well known results as

well as powerful original methods, each is explained and illustrated by

carefully selected problems.

*Chapter 1. Inequalities between means. *This chapter starts with the fundamental fact , upon which many

interesting inequalities are derived. It also serves to emphasize that powerful

results can be obtained by little means.

*Chapter 2. Cauchy-Schwars. *The classical result of Cauchy-Schwarz is revisited with examples illustrating

sophisticated, at time surprising, ways to apply the inequality. Other

classical inequalities such as those of Chebyshev and H\”older are also

discussed and shown how they might cooperate with Cauchy-Schwarz inequality.

*Chapter 3. Convexity. *This chapter utilises calculus in solving inequalities. Based on simpe

properties of linear and convex functions, systematic methods are derived

to tackle some advanced problems. Also discussed is tangent line method,

which gives a geometric interpretation of bounds.

*Chapter 4. Homogenous inequalities. *Homogeneous inequalities constitutes a large class of inequality problems.

This chapter discusses various approaches to solving this class of inequalities, including the techniques of homogenization, normalisation, the application of Rolle’s theorem to reduce the number of variables, the use of limits and partitions, quadratic estimations, and establishing new bounds through isolated fudging. Especially in focus are powerful techniques to solve inequalities by the change of variables , and and by

transforming them to one of the following forms.

$latex x(a-b)(a-c)+y(b-c)(b-a)+z(c-a)(c-b)\ge 0$,

,

.

All three, four variable symmetric polynomial inequalities can be solved using ideas in this chapter.

*Chapter 5. The method of Mixing Variables. *The method of mixing variables has been used in various forms for decades – an

example is G. Polya’s delightful proof of the AM-GM inequalities. This chapter

examines this idea in depth with extension in different directions.

The first three sections explain why mixing variables work, give hints to

find approriate variables to mix by taking equality cases into consideration.

The most important results in this chapter are two theorems which facilitates

solutions for a large class of multi-variable inequalities.

*Chapter 6. Further Topics and problems with solutions. *The chapter starts with miscellaneous indenpendent topics touching upon various aspects of solving inequalities. The discussion includes

the interplay between trogonometric and algebraic substitution,

absolute values, inequalities with special equality cases and

inequalities with ordered sequences.

For pdf file: sample-ineqs.pdf

PLEASE GOES TO BOOK

please goes to books

quyen sach do duoc ban o dau ?

Where and How can I get this book. I find it really interesting!

i live in india.how can i buy this book?