Important Olympiad-inequalities [duplicate]

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12












$begingroup$



This question already has an answer here:



  • What are the most popular techniques of proving inequalities?

    3 answers



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.










share|cite|improve this question











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marked as duplicate by Michael Rozenberg inequality
Users with the  inequality badge can single-handedly close inequality questions as duplicates and reopen them as needed.

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Feb 10 at 3:18


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    @Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
    $endgroup$
    – Dr. Mathva
    Feb 10 at 11:39










  • $begingroup$
    All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:42










  • $begingroup$
    @Michael Rozenberg What do you mean by IMC?
    $endgroup$
    – Dr. Mathva
    Feb 10 at 14:22










  • $begingroup$
    See here: imc-math.org.uk
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 14:25















12












$begingroup$



This question already has an answer here:



  • What are the most popular techniques of proving inequalities?

    3 answers



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.










share|cite|improve this question











$endgroup$



marked as duplicate by Michael Rozenberg inequality
Users with the  inequality badge can single-handedly close inequality questions as duplicates and reopen them as needed.

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Feb 10 at 3:18


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

















  • $begingroup$
    @Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
    $endgroup$
    – Dr. Mathva
    Feb 10 at 11:39










  • $begingroup$
    All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:42










  • $begingroup$
    @Michael Rozenberg What do you mean by IMC?
    $endgroup$
    – Dr. Mathva
    Feb 10 at 14:22










  • $begingroup$
    See here: imc-math.org.uk
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 14:25













12












12








12


10



$begingroup$



This question already has an answer here:



  • What are the most popular techniques of proving inequalities?

    3 answers



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.










share|cite|improve this question











$endgroup$





This question already has an answer here:



  • What are the most popular techniques of proving inequalities?

    3 answers



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.





This question already has an answer here:



  • What are the most popular techniques of proving inequalities?

    3 answers







inequality contest-math big-list






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 10 at 2:39









Alexander Gruber

20.1k25102173




20.1k25102173










asked Feb 9 at 19:24









Dr. MathvaDr. Mathva

2,114324




2,114324




marked as duplicate by Michael Rozenberg inequality
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Feb 10 at 3:18


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Michael Rozenberg inequality
Users with the  inequality badge can single-handedly close inequality questions as duplicates and reopen them as needed.

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Feb 10 at 3:18


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • $begingroup$
    @Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
    $endgroup$
    – Dr. Mathva
    Feb 10 at 11:39










  • $begingroup$
    All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:42










  • $begingroup$
    @Michael Rozenberg What do you mean by IMC?
    $endgroup$
    – Dr. Mathva
    Feb 10 at 14:22










  • $begingroup$
    See here: imc-math.org.uk
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 14:25
















  • $begingroup$
    @Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
    $endgroup$
    – Dr. Mathva
    Feb 10 at 11:39










  • $begingroup$
    All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:42










  • $begingroup$
    @Michael Rozenberg What do you mean by IMC?
    $endgroup$
    – Dr. Mathva
    Feb 10 at 14:22










  • $begingroup$
    See here: imc-math.org.uk
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 14:25















$begingroup$
@Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
$endgroup$
– Dr. Mathva
Feb 10 at 11:39




$begingroup$
@Michael Rozenberg, I know that the question you've linked might look similar to mine; however, I wanted to emphasize the fact that I'm looking for $mathbfolympiad$ inequalities, which has nothing to do with the inequalities you might require for the maths-degree for instance...
$endgroup$
– Dr. Mathva
Feb 10 at 11:39












$begingroup$
All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
$endgroup$
– Michael Rozenberg
Feb 10 at 12:42




$begingroup$
All these they are Olimpiad inequalities. I think these themes they are same. Remember, there is also IMC.
$endgroup$
– Michael Rozenberg
Feb 10 at 12:42












$begingroup$
@Michael Rozenberg What do you mean by IMC?
$endgroup$
– Dr. Mathva
Feb 10 at 14:22




$begingroup$
@Michael Rozenberg What do you mean by IMC?
$endgroup$
– Dr. Mathva
Feb 10 at 14:22












$begingroup$
See here: imc-math.org.uk
$endgroup$
– Michael Rozenberg
Feb 10 at 14:25




$begingroup$
See here: imc-math.org.uk
$endgroup$
– Michael Rozenberg
Feb 10 at 14:25










2 Answers
2






active

oldest

votes


















12












$begingroup$

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":



enter image description here



enter image description here



enter image description here



enter image description here



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



enter image description here



The book covers in extensive detail the following topics:



enter image description here



enter image description here



Also a fine reading:



A Brief Introduction to Olympiad Inequalities, Evan Chen






share|cite|improve this answer











$endgroup$








  • 2




    $begingroup$
    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.
    $endgroup$
    – Xander Henderson
    Feb 11 at 18:00


















5












$begingroup$

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:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abcgeq0.$$



Proof:



It's $$Rgeq2r,$$ 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.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Similar to this, which was also written by OP
    $endgroup$
    – user574848
    Feb 10 at 1:54










  • $begingroup$
    This is the link. Thank you!
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 3:17










  • $begingroup$
    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
    $endgroup$
    – stressed out
    Feb 10 at 12:16











  • $begingroup$
    @stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:52







  • 1




    $begingroup$
    @stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:58


















2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









12












$begingroup$

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":



enter image description here



enter image description here



enter image description here



enter image description here



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



enter image description here



The book covers in extensive detail the following topics:



enter image description here



enter image description here



Also a fine reading:



A Brief Introduction to Olympiad Inequalities, Evan Chen






share|cite|improve this answer











$endgroup$








  • 2




    $begingroup$
    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.
    $endgroup$
    – Xander Henderson
    Feb 11 at 18:00















12












$begingroup$

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":



enter image description here



enter image description here



enter image description here



enter image description here



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



enter image description here



The book covers in extensive detail the following topics:



enter image description here



enter image description here



Also a fine reading:



A Brief Introduction to Olympiad Inequalities, Evan Chen






share|cite|improve this answer











$endgroup$








  • 2




    $begingroup$
    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.
    $endgroup$
    – Xander Henderson
    Feb 11 at 18:00













12












12








12





$begingroup$

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":



enter image description here



enter image description here



enter image description here



enter image description here



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



enter image description here



The book covers in extensive detail the following topics:



enter image description here



enter image description here



Also a fine reading:



A Brief Introduction to Olympiad Inequalities, Evan Chen






share|cite|improve this answer











$endgroup$



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":



enter image description here



enter image description here



enter image description here



enter image description here



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



enter image description here



The book covers in extensive detail the following topics:



enter image description here



enter image description here



Also a fine reading:



A Brief Introduction to Olympiad Inequalities, Evan Chen







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Feb 10 at 19:31

























answered Feb 9 at 19:48









OldboyOldboy

8,62011036




8,62011036







  • 2




    $begingroup$
    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.
    $endgroup$
    – Xander Henderson
    Feb 11 at 18:00












  • 2




    $begingroup$
    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.
    $endgroup$
    – Xander Henderson
    Feb 11 at 18:00







2




2




$begingroup$
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.
$endgroup$
– Xander Henderson
Feb 11 at 18:00




$begingroup$
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.
$endgroup$
– Xander Henderson
Feb 11 at 18:00











5












$begingroup$

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:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abcgeq0.$$



Proof:



It's $$Rgeq2r,$$ 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.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Similar to this, which was also written by OP
    $endgroup$
    – user574848
    Feb 10 at 1:54










  • $begingroup$
    This is the link. Thank you!
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 3:17










  • $begingroup$
    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
    $endgroup$
    – stressed out
    Feb 10 at 12:16











  • $begingroup$
    @stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:52







  • 1




    $begingroup$
    @stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:58
















5












$begingroup$

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:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abcgeq0.$$



Proof:



It's $$Rgeq2r,$$ 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.






share|cite|improve this answer











$endgroup$












  • $begingroup$
    Similar to this, which was also written by OP
    $endgroup$
    – user574848
    Feb 10 at 1:54










  • $begingroup$
    This is the link. Thank you!
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 3:17










  • $begingroup$
    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
    $endgroup$
    – stressed out
    Feb 10 at 12:16











  • $begingroup$
    @stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:52







  • 1




    $begingroup$
    @stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:58














5












5








5





$begingroup$

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:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abcgeq0.$$



Proof:



It's $$Rgeq2r,$$ 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.






share|cite|improve this answer











$endgroup$



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:
$$a^3+b^3+c^3-a^2b-a^2c-b^2a-b^2c-c^2a-c^2b+3abcgeq0.$$



Proof:



It's $$Rgeq2r,$$ 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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Feb 10 at 12:51

























answered Feb 9 at 20:50









Michael RozenbergMichael Rozenberg

106k1894198




106k1894198











  • $begingroup$
    Similar to this, which was also written by OP
    $endgroup$
    – user574848
    Feb 10 at 1:54










  • $begingroup$
    This is the link. Thank you!
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 3:17










  • $begingroup$
    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
    $endgroup$
    – stressed out
    Feb 10 at 12:16











  • $begingroup$
    @stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:52







  • 1




    $begingroup$
    @stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:58

















  • $begingroup$
    Similar to this, which was also written by OP
    $endgroup$
    – user574848
    Feb 10 at 1:54










  • $begingroup$
    This is the link. Thank you!
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 3:17










  • $begingroup$
    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
    $endgroup$
    – stressed out
    Feb 10 at 12:16











  • $begingroup$
    @stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:52







  • 1




    $begingroup$
    @stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
    $endgroup$
    – Michael Rozenberg
    Feb 10 at 12:58
















$begingroup$
Similar to this, which was also written by OP
$endgroup$
– user574848
Feb 10 at 1:54




$begingroup$
Similar to this, which was also written by OP
$endgroup$
– user574848
Feb 10 at 1:54












$begingroup$
This is the link. Thank you!
$endgroup$
– Michael Rozenberg
Feb 10 at 3:17




$begingroup$
This is the link. Thank you!
$endgroup$
– Michael Rozenberg
Feb 10 at 3:17












$begingroup$
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
$endgroup$
– stressed out
Feb 10 at 12:16





$begingroup$
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
$endgroup$
– stressed out
Feb 10 at 12:16













$begingroup$
@stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
$endgroup$
– Michael Rozenberg
Feb 10 at 12:52





$begingroup$
@stressed out I added something. See now. About Bacteria see here: math.stackexchange.com/questions/2903914
$endgroup$
– Michael Rozenberg
Feb 10 at 12:52





1




1




$begingroup$
@stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
$endgroup$
– Michael Rozenberg
Feb 10 at 12:58





$begingroup$
@stressed out Yes, of course! But it's a semi-perimeter. Sometimes it's very useful.
$endgroup$
– Michael Rozenberg
Feb 10 at 12:58



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