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 nd
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.
Finally, though this note is aimed for students participating in the Bangladesh Mathematical Olympiad
who will be hoping to be in the Bangladesh IMO team I hope it will be useful for everyone. I am really
grateful to the MathLinks forum for supplying me with the huge collection of problems.
Samin Riasat
28 November, 2008
http://ifile.it/l3iv74a/basics.pdf
