Algebra
# Power Mean Inequalities

\[A = \dfrac{12 x^{3} y + 108x y^{3} + 81y^{4} + x^{4} + 1 + 2y + y^{2} + 54x^{2} y^{2} }{x^{2} + x^{2} y + 6xy + 9y^{3} + 6xy^{2} + 9y^{2}}\]

Let \(T\) be the minimum value of \(A\) for positive real numbers \(x,y\). Find the value of \(5T\).

Let \(0 < x \leq y \leq z\) such that \(xy+yz+zx=3\).

Find the maximum value of \(xy^3z^2\).

\[\frac { x }{ y } +\frac { y }{ z+1 } +\frac { z }{ x } =\frac { 5 }{ 2 }\]

How many ordered triples of positive integers \((x,y,z) \) satisfy the equation above?

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 \(A^{2}\).

If \(a\) and \(b\) are positive numbers, find the maximum value of \(ab(72-3a-4b) \).

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