Samin Riasat. Basics of Olympiad Inequalities

Bangladesh, 2008.The aim of this note is to acquaint students, who want to participate in mathematical Olympiads, to Olympiad level inequalities from the basics. Inequalities are used in all fields of mathematics. They have some very interesting properties and numerous applications. Inequalities are often hard to solve, and it is not always possible to find a nice solution. But it is worth approaching an inequality rather than solving it. Most inequalities need to be transformed into a suitable form by algebraic means before applying some theorem. This is what makes the problem rather difficult. Throughout this little note you will find different ways and approaches to solve an inequality. Most of the problems are recent and thus need a fruitful combination of wisely applied techniques. It took me around two years to complete this; although I didn’t work on it for some months during this period. I have tried to demonstrate how one can use the classical inequalities through different examples that show different ways of applying them. After almost each section there are some exercise problems for the reader to get his/her hands dirty! And at the end of each chapter some harder problems are given for those looking for challenges. Some additional exercises are given at the end of the book for the reader to practice his/her skills. Solutions to some selected problems are given in the last chapter to present different strategies and techniques of solving inequality problems. In conclusion, I have tried to explain that inequalities can be overcome through practice and more practice.The AM-GM Inequality
General AM-GM Inequality
Weighted AM-GM Inequality
More Challenging Problems
Cauchy-Schwarz and Holder’s Inequalities
Cauchy-Schwarz Inequality
Holder’s Inequality
More Challenging Problems
Rearrangement and Chebyshev’s Inequalities
Rearrangement Inequality
Chebyshev’s inequality.
More Challenging Problems
Other Useful Strategies
Schur’s Inequality
Jensen’s Inequality
Minkowski’s Inequality
Ravi Transformation
Normalization
Homogenization
Supplementary Problems
Hints and Solutions to Selected Problems