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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 3 Challenges

$\large \dfrac{x^2+2-\sqrt{x^4+4}}{x}$ For real $$x$$, find the closed form of the maximum value of the expression above.

Note: Try to solve this problem without using calculus.

$$a$$, $$b$$, and $$c$$ are real numbers such that

$a+b=6 \\ a^2+b^2+c^2=18$

What is the value of $$a-c$$?

For some positive reals satisfying $$\displaystyle{ \sum_{i=1}^{24} x_i=1}$$, determine the maximum possible value of $\displaystyle{ \left (\sum_{i=1}^{24}\sqrt{x_i} \right ) \left (\sum_{i=1}^{24} \frac{1}{\sqrt{1+x_i}} \right )}.$

$\large (1 + a)(1 + b)(1 + c)(1 + d)$

If $$a,b,c$$ and $$d$$ are four positive real numbers such that $$a\times b\times c\times d = 1$$, then find the minimum value of the expression above.

For $$a,b,c>0$$ and $$a+b+c=6$$. Find the minimum value of

$\large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2}$

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