AM/GM
Intermediate
Let $x, y, z$ be strictly positive real numbers. Prove that $$\Big(\frac{x}{y}+\frac{z}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{y}{z}+\frac{x}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{z}{x}+\frac{y}{\sqrt[3]{xyz}}\Big)^2 \ge 12$$
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