Challenge Problem #7: Σ(x/(y^3+2))≥1

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The forum discussion centers on proving the inequality $$\frac{x}{y^3+2}+\frac{y}{z^3+2}+\frac{z}{x^3+2} \geqslant 1$$ for positive real numbers \(x, y, z\) such that \(xyz=1\). Participants emphasize the importance of applying the AM-GM inequality and suggest exploring symmetric properties in the variables. The discussion highlights the necessity of understanding inequalities in the context of constrained variables.

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  • Understanding of the AM-GM inequality
  • Familiarity with symmetric inequalities
  • Knowledge of positive real number properties
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  • Study advanced applications of the AM-GM inequality
  • Explore symmetric functions and their properties
  • Research techniques for proving inequalities in constrained variables
  • Practice solving similar inequality problems in mathematical competitions
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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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