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Algebra

# Power Mean Inequalities: Level 4 Challenges

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

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 $$A^{2}$$.

###### This problem is inspired by Joel Tan.

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

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