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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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