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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.

Let \(x, y, z\) be real numbers such that \(x^{2}+y^{2}+z^{2}=1\). Let the maximum possible value of **\(\sqrt {6} xy+4yz\)** be \(A\).Find \(A^{2}\).

Consider all positive reals \(a\) and \(b\) such that

\[ ab (a+b) = 2000. \]

What is the minimum value of

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{a+b} ?\]

Let \(x, y, z\geq 0\) be reals such that \(x+y+z=1\). Find the maximum possible value of

\[x (x+y)^{2}(y+z)^{3}(x+z)^{4}.\]

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