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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 3

         

\[\large \dfrac{x^2+2-\sqrt{x^4+4}}{x}\] For real \(x\), find the closed form of the maximum value of the expression above.
Give your answer to 3 decimal places.

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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