Important Olympiad-inequalities [duplicate]

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

I did not find a link, but I wrote about this theme already.

I'll write something again.

There are many methods:

1. Cauchy-Schwarz (C-S)

2. AM-GM

3. Holder

4. Jensen

5. Minkowski

6. Maclaurin

7. Rearrangement

8. Chebyshov

9. Muirhead

10. Karamata

11. Lagrange multipliers

12. Buffalo Way (BW)

13. Contradiction

14. Tangent Line method

15. Schur

16. Sum Of Squares (SOS)

17. Schur-SOS method (S-S)

18. Bernoulli

19. Bacteria

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

21. E-V Method by V.Cirtoaje

22. uvw

23. Inequalities like Schur

24. 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.