MHB Challenge Problem #7: Σ(x/(y^3+2))≥1

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The discussion revolves around proving the inequality Σ(x/(y^3+2))≥1 for positive real numbers x, y, and z such that xyz=1. Participants are encouraged to explore various mathematical approaches to demonstrate the validity of the inequality. The conversation highlights the importance of manipulating the terms and applying known inequalities to achieve the proof. A hint is provided to guide the participants in their reasoning. Engaging with this problem can deepen understanding of inequalities in mathematical analysis.
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Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$
 
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A hint ...

is requested (Giggle)
 
lfdahl said:
A hint ...

is requested (Giggle)
Niiiiice. (Clapping)

-Dan
 

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