Prove that $(a b + x y) (a x + b y) > 4 a b x y$ .

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Solution

This is only true if $a, b, x, y$ are positive numbers
Since $a, b, x, y$ are positive $a b, x y, a x, b y$ are also positive
Apply $A M \geq G M$ for $a b, x y$
$\Rightarrow \frac{a b + x y}{2} \geq \sqrt{a b x y} \to$ eqn 1
Apply $A M \geq G M$ for $a x, b y$
$\frac{a x + b y}{2} \geq \sqrt{a b x y} \to$ eqn 2
Multiplying the corresponding sides of eqn 1 and eqn 2
$\Rightarrow (a b + x y) (a x + b y) \geq 4 a b x y$
Hence proved

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