Important Olympiad-inequalities 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)

*Cauchy-Schwarz

*Jensen
...
Could you suggest some more?

This question was inspired by the fantastic contributions of @Michael Rozenberg on inequalities. 
 A: I did not find a link, but I wrote about this theme already. 
I'll write something again.
There are many methods:


*

*Cauchy-Schwarz (C-S)

*AM-GM

*Holder

*Jensen

*Minkowski

*Maclaurin

*Rearrangement

*Chebyshov

*Muirhead

*Karamata

*Lagrange multipliers 

*Buffalo Way (BW)

*Contradiction 

*Tangent Line method

*Schur

*Sum Of Squares (SOS)

*Schur-SOS method (S-S)

*Bernoulli 

*Bacteria 

*RCF, LCF, HCF (with half convex, half concave functions) by V.Cirtoaje

*E-V Method by  V.Cirtoaje

*uvw

*Inequalities like Schur

*pRr method for the geometric inequalities
and more.
In my opinion, the best book it's the inequalities forum in the AoPS: https://artofproblemsolving.com/community/c6t243f6_inequalities
Just read it!
Also, there is the last book by Vasile Cirtoaje (2018) and his papers. 
An example for using pRr.
Let $a$, $b$ and $c$ be sides-lengths of a triangle. Prove that:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abc\geq0.$$
Proof:
It's $$R\geq2r,$$ which is obvious.
Actually, the inequality $$\sum_{cyc}(a^3-a^2b-a^2c+abc)\geq0$$ is true for all non-negatives $a$, $b$ and $c$ and named as the Schur's inequality. 
A: Essential reading:
Olympiad Inequalities, Thomas J. Mildorf
All useful inequalities are clearly listed and explaind on the first few pages. Mildorf calls them "The Standard Dozen":




EDIT: If you look for a good book, here is my favorite one:

The book covers in extensive detail the following topics:


Also a fine reading:
A Brief Introduction to Olympiad Inequalities, Evan Chen
