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About the notes
This set of notes is intended for people who want to learn inequalities for pre-university international competitions (e.g. International Mathematical Olympiad).
Inequalities is amongst the hardest topics in mathematical competitions because a distinct set of thinking skills is required to solve the problems. It is almost impossible to divide the problem into simpler parts, and therefore mathematical induction is usually ineffective. (One of my teammates has attempted M.I. for IMO 1997 #3, and has gotten a 0/7 score).
For each problem, the equality case(s), if any, should be given with the proof of each inequality.
New learners should read all examples before attempting any problems in the corresponding chapter. The exercises are hard - be prepared to spend more than 15 minutes for the easiest questions. For the harder questions, days or even months are needed.
As you get experienced, you will be able to get a proof in a much shorter time.