Important Olympiad-inequalities

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":

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

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The book covers in extensive detail the following topics:

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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.

Tags:

Inequality

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