By Dr. Mathva


2019-02-09 19:24:23 8 Comments

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?


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

2 comments

@Oldboy 2019-02-09 19:48:27

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

@Xander Henderson 2019-02-11 18:00:32

While the reference you provide might be useful, displaying text-as-images makes your answer inaccessible to screenreaders, and hampers the searchability of your answer. It would be preferable if you could summarize (in text) the major important results.

@Michael Rozenberg 2019-02-09 20:50:44

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.

@user574848 2019-02-10 01:54:55

Similar to this, which was also written by OP

@Michael Rozenberg 2019-02-10 03:17:33

This is the link. Thank you!

@stressed out 2019-02-10 12:16:08

Not that it's very important, but you're missing a dot after 16 which has kind of given a bad look to the list. :P Also, what is the pRr method? I googled it but ended up with results in biology which I don't think are very relevant. It's hard to find relevant results about some of the acronyms you used on Google. Don't even get me started on "Bacteria". :P

@Michael Rozenberg 2019-02-10 12:52:17

@stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914

@stressed out 2019-02-10 12:56:47

Thank you. So, pRr method is about the relations among the perimeter, inscribed and circumscribed circles of a triangle?

@Michael Rozenberg 2019-02-10 12:58:40

@stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.

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