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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 \(A^{2}\).

\[ \large { x }^{ 6 }-12{ x }^{ 5 }+a{ x }^{ 4 }+b{ x }^{ 3 }+c{ x }^{ 2 }+dx+64 =0 \]

Let \(a,b,c,d\) be all constants.

If all of the roots of the above equation are positive, find \(b + c - a\).

For positive real numbers \( a, b, c \), find the minimum integer value possible of the following equation:

\[ 6a^{3} + 9b^{3} + 32c^{3} + \frac{1}{4abc} \]

Hint: Click here for hint.

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