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Power Mean Inequalities

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.

Level 2

         

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

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

Given that \(a,\) \(b,\) and \(c\) are positive reals such that \(a+b+c=6\), find the maximum possible value of \(abc\).

Find the minimum value of \({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }\) for positive numbers \(a,b,c\) satisfying the constraint \(a+b+c=6\).

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

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

Let \(x,y\) be positive reals such that \(x+y =2\), find the maximum value of \(x^{3}y^{3}(x^{3}+y^{3})\).

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