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Question Number 151142 by mathdanisur last updated on 18/Aug/21

$$\mathrm{if}\mathrm{x};\mathrm{y};\mathrm{z}\in {\mathbb{R}}^{+}\mathrm{and}\frac{1}{{\mathrm{x}}^2}+\frac{1}{{\mathrm{y}}^2}+\frac{1}{{\mathrm{z}}^2}=\frac{27}{4}$$$$\mathrm{prove}\mathrm{that}:$$$$\frac{{\mathrm{x}}^3+{\mathrm{y}}^2}{{\mathrm{x}}^2+{\mathrm{y}}^2}+\frac{{\mathrm{y}}^3+{\mathrm{z}}^2}{{\mathrm{y}}^2+{\mathrm{z}}^2}+\frac{{\mathrm{z}}^3+{\mathrm{x}}^2}{{\mathrm{z}}^2+{\mathrm{x}}^2}⩾\frac{5}{2}$$

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