Prove that $\displaystyle \frac 1x + \frac 1y + \frac 1z = 1\Rightarrow 8 + (z - 1)\left((x - 1)(y - 1)\right)^2\geq \frac {4xy}{z^2} + 4xy$ $\displaystyle x, y, z \in \mathbb{R^+}$
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