Important Olympiad-inequalities

Question

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

1. AM-GM (and the weighted one)

2. Cauchy-Schwarz

3. Jensen

...

Could you suggest some more?

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This question was inspired by the fantastic contributions of @Michael Rozenberg on inequalities.

Answer 1

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

Answer 2

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.