TITLE: Important Olympiad-inequalities
QUESTION [15 upvotes]: 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.

REPLY [7 votes]: 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.