Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. Which of the following statements are true?

$I$. $4$ divides $x$ or $4$ divides $y$.

$II$. $3$ divides $x+y$ or $3$ divides $x-y$.

$III$. $5$ divides $z\left(x^2-y^2\right)$.

Let $a$ , $b$ , $c$ are roots of equation $x^3 + 8x + 1 = 0$ ,then the value of 

 $\frac{{bc}}{{(8b + 1)(8c + 1)}} + \frac{{ac}}{{(8a + 1)(8c + 1)}} + \frac{{ab}}{{(8a + 1)(8b + 1)}}$ is equal to

$\{ x \in R:|x - 2|\,\, = {x^2}\} = $

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is

Let $a, b, c, d$ be numbers in the set $\{1,2,3,4,5,6\}$ such that the curves $y=2 x^3+a x+b$ and $y=2 x^3+c x+d$ have no point in common. The maximum possible value of $(a-c)^2+b-d$ is

The roots of the equation ${x^4} - 2{x^3} + x = 380$ are