Tricky inequality no avail to AM-GMLet a , b , c be 3 distinct positive real numbers such that abc = 1. Prove that a 3 ( a − b ) ( a − c ) + b 3 ( b − c ) ( b − a ) + c 3 ( c − a ) ( c − b ) ≥ 3I tried AM-GM in many different ways, but it doesn't work since one of the terms on the LHS inevitably becomes negative. Any help is greatly appreciated.

Question

Tricky inequality no avail to AM-GMLet   a  ,  b  ,  c be 3 distinct positive real numbers such that abc = 1. Prove that            a      3                      (        a        −        b        )                    (        a        −        c        )                 +            b      3                      (        b        −        c        )                    (        b        −        a        )                 +               c      3                      (        c        −        a        )                    (        c        −        b        )                 ≥  3I tried AM-GM in many different ways, but it doesn&#039;t work since one of the terms on the LHS inevitably becomes negative. Any help is greatly appreciated.

Jordan Mcpherson · Accepted Answer

By AM-GM      ∑          c      y      c                  a      3              (      a      −      b      )      (      a      −      c      )        =  −      ∑          c      y      c                  a      3              (      a      −      b      )      (      c      −      a      )        =  −      ∑          c      y      c                          a        3            (      b      −      c      )                      ∏                  c          y          c                    (      a      −      b      )        =  a  +  b  +  c  ≥  3

Tricky inequality no avail to AM-GM Let a , b , c be 3 distinct positive real nu

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