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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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