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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

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.

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

$\large \frac{1}{\sqrt{5x+1}}+\frac{1}{\sqrt{x^2-3x+4}}\geq\frac{x+3}{2(x+1)}$ How many integer $$x$$ satisfy the inequality above

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.

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

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