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This is the task to find the minimum value of $x^5 + y^5 + z^5 - 5xyz$ where $x,y$ and $z$ are any positive numbers.

I know that I may use this inequality: $$(t_1\cdot t_2\cdot t_3\cdots t_n)^{\frac{1}{n}} \leq \frac{t_1+t_2+t_3+\cdots +t_n}{n}$$

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By the inequality we have that $\frac{x^5 + y^5 + z^5 + 1 + 1}{5} \ge \sqrt[5]{x^5y^5z^5\cdot1 \cdot 1}$. Hence $x^5 + y^5 + z^5 + 2 \ge 5xyz$

$$x^5 + y^5 + z^5 - 5xyz = x^5 + y^5 + z^5 + 2 - 5xyz - 2 \ge 5xyz - 5xyz - 2 = -2$$

It's obtained for $x=y=z$

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Let $xyz=t^3$.

Hence, by AM-GM (which is just Jensen!) $x^5+y^5+z^5-5xyz\geq3t^5-5t^3\geq-2,$

where the last inequality it's AM-GM again: $3t^5+2\geq5\sqrt[5]{t^{15}}=5t^3$

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