This is a problem from RMO 2015 Mumbai Region based on inequality.
Problem: RMO 2015 Mumbai Region
Let x, y, z be real numbers such that . Prove that
Note that .
According to the given condition .
Adding 2(x+y+z) + 1 to both sides,
We wish to show
Replacing 2xyz by we have
(this is what we need to show).
Therefore we need to show
This is true by Cauchy Schwarz Inequality.
PROOF 2 (suggested by Arkabrata Das)
CHECK FILE: new doc 1720151229235503023