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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.

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 \(A^{2}\).

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Given that \(x\), \(y\), and \(z\) are positive real numbers satisfying \(xyz(x+y+z)=1\), minimize \((x+y)(y+z)(z+x)\).

Enter your answer to five decimal places.

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Find the minimum value of \(x^6 + y^6 -54xy \), where \(x\) and \(y\) are real numbers.

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