Let $x,y,z$ be positive reals. Which of the following implies $x=y=z?$

I. $x^3+y^3+z^3=3xyz$

II. $x^3+y^{2}z+y{z}^2=3xyz$

II. $x^3+y^{2}z+z^{2}x=3xyz$

IV. ${\left(x+y+z\right)}^3=27xyz$

Let $n\ge 4$ be a positive integer and let $l_1,l_2,\dots ,l_n$ be the lengths of the sides of arbitrary $n$ sided non-degenerate polygon $P$. Suppose

$\frac{l_1}{l_2}+\frac{l_2}{l_3}+\dots +\frac{l_{n-1}}{l_n}+\frac{l_n}{l_1}=n$

Consider the following statements: 

$\mathrm{I}$. The lengths of the sides of $P$ are equal.
$\mathrm{II}$. The angles of $P$ are equal.
$\mathrm{III}$. $P$ is a regular polygon if it is cyclic. Then,

If the arithmetic mean of two numbers $a$ and $b, a>b>0$ , is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to: