work 020 – IMOTC B.J.Venkatachala

For x,y,z are positive reals.
Prove that 

\dfrac{1}{2} \sum \dfrac{1}{xy}\ge \sum \dfrac{1}{x^2 + yz}.
Proof 
Let ,
xy = \frac{1}{a^2}

yz = \frac {1}{b^2}

zx = \frac {1}{c^2}
Thus ,
x^2 = \frac {b^2}{a^2c^2} \hdots
substituting in the question we have –
LHS = \frac {1}{2} \cdot [a^2 + b^2 + c^2]
and,
RHS = \sum \frac {a^2b^2c^2}{b^4 + a^2c^2}
By AM-GM ,
RHS \le \sum \frac {b^2a^2c^2}{2b^2ac} = \sum \frac {ac}{2}

$latex \sum \frac {ac}{2} \le \sum \frac {a^2}{2} = LHS$
and we are done.
With equality for a = b = c \implies x = y = z

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