Power Mean Inequalities: Level 4 Challenges Practice Problems Online

Find the minimum value of $x^6 + y^6 -54xy$ , where $x$ and $y$ are real numbers.

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:

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

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

Power Mean Inequalities: Level 4 Challenges