[mkup]

Problem 5 has a simpler geometric solution, it's enough to plot circle of radius sqrt(2) on the Oab plane and a few lines b=-a, b=-a+1, b=-a+2, b=-a-1, b=-a-2 to observe the intersections. The rest of the lines are too far from this circle to intersect with it.

[nomemory]

That's cool observation.

It's a good reflex when you see the sum of two squares to think at the Circle.

[lormayna]

Problem 6 has a similar resolution too: you have the circle of radius sqrt(3) and the rect a+b=n^2.

Considering the fact that n is natural is easy to find the solutions: n=0 and n=1