QM-AM-GM-HM Practice Problems Online

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?

f(m) \geq f(n)

f(n) \geq f(m)

f(m) = f(n)

None of the above

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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Power Mean Inequalities

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