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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 \((x+y)(y+z)(z+x)\).

Enter your answer to five decimal places.

\[\large \frac{1}{\sqrt{5x+1}}+\frac{1}{\sqrt{x^2-3x+4}}\geq\frac{x+3}{2(x+1)}\] 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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