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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.

# Level 4

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

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

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

$$\alpha, \beta > 0$$.

With $$\alpha, \beta > 0$$, if $$\alpha + \dfrac{1}{\alpha}$$ and $$2 - \beta - \dfrac{1}{\beta}$$ are the roots of the quadratic equation

$x^{2} -2(a+1)x + a - 3 = 0,$

then find the sum of integral values of $$a$$.

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

$x (x+y)^{2}(y+z)^{3}(x+z)^{4}.$

Given that $$a,b,c$$ are non-negative real numbers, then $ab^2+2bc^2+3ca^2\ge kabc$ for some positive real $$k$$. What is the largest possible value of $$k$$? Round to the nearest thousandth.

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