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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.

# AM-GM

Suppose the average of $$5$$ positive numbers is equal to $$2.$$ What is the maximum possible product of the numbers?

Suppose $$xyz = 8$$ with $$x, y, z > 0.$$ What is the minimum value of $$x + y + z$$?

What is the largest $$n$$ for which

$x^3 + y^3 + z^3 \geq nxyz$

holds for all $$x, y, z > 0$$?

What is the largest $$n$$ for which

$(x + y)(y + z)(x + z) \geq nxyz$

holds for all positive $$x, y, z$$?

Suppose $$2x + 3y + z = 18.$$ Find the maximum value of $$xyz$$ for $$x, y, z > 0.$$

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