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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.

# QM-AM-GM-HM

Given positive $$a_1, a_2, \ldots, a_j,$$ let $$f$$ be defined as

$f(n) = \left(\frac{a_1^n + a_2^n + \cdots + a_j^n}{j}\right)^{1/n}.$

For $$m > n$$, which of the following is necessarily true?

If the arithmetic mean and quadratic mean (root mean square) of a set of positive numbers are equal, which of the following means must also equal that same value?

What is the smallest $$n$$ for which

$\left(a^3 + b^3\right)^2 \leq n (a^6 + b^6)$

holds for all positive $$a$$ and $$b$$?

What is the largest $$n$$ for which

$(x + y) (y + z) (x + y + z) \geq \frac{n x^2 y^3 z^2}{(x + y)(y + z)(xy + xz + yz)}$

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

Suppose the harmonic mean of $$5$$ positive numbers is equal to $$2.$$ What is the minimum possible sum of the numbers?

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