JBMO 2005

Find all positive integers $x,y$ satisfying the equation $$9(x^2+y^2+1) + 2(3xy+2) = 2005 .$$

Prove that there exist

(a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones;

(b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.

Find all 3-digit positive integers $\overline{abc}$ such that $$\overline{abc} = abc(a+b+c) ,$$ where $\overline{abc}$ is the decimal representation of the number.