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This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

\[\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a} \]

If \( a, b, c \) and \( d \) are any four positive real numbers, then find the minimum value of the expression above.

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For \(a,b,c>0\) and \(a+b+c=6\). Find the minimum value of

\[ \large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2} \]

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