As an olympiad-participant, I've had to solve numerous inequalities; some easy ones and some very difficult ones. Inequalities might appear in every Olympiad discipline (Number theory, Algebra, Geometry and Combinatorics) and usually require previous manipulations, which makes them even harder to solve...
Some time ago, someone told me that
Solving inequalities is kind of applying the same hundred tricks again and again
And in fact, knowledge and experience play a fundamental role when it comes to proving/solving inequalities, rather than instinct.
This is the reason why I wanted to gather the most important Olympiad-inequalities such as
AM-GM (and the weighted one)
Could you suggest some more?
This question was inspired by the fantastic contributions of @Michael Rozenberg on inequalities.