Tricky inequality no avail to AM-GMLet a , b , c be 3 distinct positive...

doturitip9

doturitip9

Answered

2022-07-07

Tricky inequality no avail to AM-GM
Let 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 )   3
I 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.

Answer & Explanation

Jordan Mcpherson

Jordan Mcpherson

Expert

2022-07-08Added 16 answers

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

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