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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.

Power Mean Inequalities: Level 4 Challenges

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.

Consider all positive reals $$a$$ and $$b$$ such that

$ab (a+b) = 2000.$

What is the minimum value of

$\frac{1}{a} + \frac{1}{b} + \frac{1}{a+b} ?$

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

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

The minimum value of $$\dfrac{(x^4+1)(y^4+1)(z^4+1)}{xy^2z}$$ as $$x,y,$$ and $$z$$ range over the positive reals is equal to $$\dfrac{A\sqrt{B}}{C},$$ where $$A$$ and $$C$$ are coprime and $$B$$ is squarefree. What is $$A+B+C?$$

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

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