Solution Set

Part I

$(9,1)(1,9)$
$9$
$484$
$(8,8)(8,-8)(-8,8)(-8,-8)$
$1+......$
$36$
$0$
$\left(\frac{9}{2},\frac{9}{4},\frac{27}{4}\right)(-36,-18,0)$
$39$
$1498$

Part II

$6$
$80$
$\frac{1}{2025}$
$(2+i,24+i)\text{ or }(24+i,2+i)$
$k\pi +\frac{\pi }{12}$ $k\pi +\frac{5\pi }{12}$ $k\pi -\frac{1}{2}arcsin(3\sqrt{3}-5)$ $k\pi +\frac{1}{2}arcsin(3\sqrt{3}-5)+\frac{\pi }{2}$
$x=y=-15$
$60$
$8\text{ or }\frac{3}{2}\sqrt{33}-4$
$924$