Algebra
# Power Mean Inequalities

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 $A^{2}$.

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.

$a, b$ and $c$ are positive real numbers greater than or equal to 1 satisfying

$\begin{aligned} abc & =100,\\ a^{\lg a} b^{\lg b} c^{\lg c} & \geq 10000. \\ \end{aligned}$

What is the value of $a + b + c$?

**Details and assumptions**:

- $\lg$ refers to log base 10.

For positive real numbers $a, b, c$, find the minimum integer value possible of the following equation:

$6a^{3} + 9b^{3} + 32c^{3} + \frac{1}{4abc}$

Hint: Click here for hint.