Important Olympiad-inequalities












11












$begingroup$


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.






inequality soft-question contest-math big-list







share|cite|improve this question









$endgroup$

















    11












    $begingroup$


    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.






    inequality soft-question contest-math big-list







    share|cite|improve this question









    $endgroup$















      11












      11








      11


      6



      $begingroup$


      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.






      inequality soft-question contest-math big-list







      share|cite|improve this question









      $endgroup$




      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.






      inequality soft-question contest-math big-list




      inequality soft-question contest-math big-list






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 7 hours ago









      Dr. MathvaDr. Mathva

      1,200318




      1,200318






















          2 Answers
          2






          active

          oldest

          votes


















          11












          $begingroup$

          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



          Also a fine reading:



          A Brief Introduction to Olympiad Inequalities, Evan Chen






          share|cite|improve this answer











          $endgroup$





















            3












            $begingroup$

            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)




            1. Schur-SOS method (S-S)


            2. Bernoulli


            3. Bacteria


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


            5. E-V Method by V.Cirtoaje


            6. uvw


            7. Inequalities like Schur


            8. 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.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Similar to this, which was also written by OP
              $endgroup$
              – user574848
              41 mins ago











            Your Answer





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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            11












            $begingroup$

            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



            Also a fine reading:



            A Brief Introduction to Olympiad Inequalities, Evan Chen






            share|cite|improve this answer











            $endgroup$


















              11












              $begingroup$

              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



              Also a fine reading:



              A Brief Introduction to Olympiad Inequalities, Evan Chen






              share|cite|improve this answer











              $endgroup$
















                11












                11








                11





                $begingroup$

                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



                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



                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 6 hours ago

























                answered 6 hours ago









                OldboyOldboy

                8,1551936




                8,1551936























                    3












                    $begingroup$

                    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)




                    1. Schur-SOS method (S-S)


                    2. Bernoulli


                    3. Bacteria


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


                    5. E-V Method by V.Cirtoaje


                    6. uvw


                    7. Inequalities like Schur


                    8. 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.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Similar to this, which was also written by OP
                      $endgroup$
                      – user574848
                      41 mins ago
















                    3












                    $begingroup$

                    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)




                    1. Schur-SOS method (S-S)


                    2. Bernoulli


                    3. Bacteria


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


                    5. E-V Method by V.Cirtoaje


                    6. uvw


                    7. Inequalities like Schur


                    8. 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.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Similar to this, which was also written by OP
                      $endgroup$
                      – user574848
                      41 mins ago














                    3












                    3








                    3





                    $begingroup$

                    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)




                    1. Schur-SOS method (S-S)


                    2. Bernoulli


                    3. Bacteria


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


                    5. E-V Method by V.Cirtoaje


                    6. uvw


                    7. Inequalities like Schur


                    8. 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.






                    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)




                    1. Schur-SOS method (S-S)


                    2. Bernoulli


                    3. Bacteria


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


                    5. E-V Method by V.Cirtoaje


                    6. uvw


                    7. Inequalities like Schur


                    8. 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.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 5 hours ago









                    Michael RozenbergMichael Rozenberg

                    103k1891195




                    103k1891195












                    • $begingroup$
                      Similar to this, which was also written by OP
                      $endgroup$
                      – user574848
                      41 mins ago


















                    • $begingroup$
                      Similar to this, which was also written by OP
                      $endgroup$
                      – user574848
                      41 mins ago
















                    $begingroup$
                    Similar to this, which was also written by OP
                    $endgroup$
                    – user574848
                    41 mins ago




                    $begingroup$
                    Similar to this, which was also written by OP
                    $endgroup$
                    – user574848
                    41 mins ago


















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