OJ Board

Any hints for this one? It reduces to counting the number of lattice points in a triangle in 3-D space. How do you do that? Is there a generalisation of Pick's theorem which could be applied here?

Short answer is:
No, there is no known generalization of Pick's theorem for 3d space.

You probably know that the given triangle lies on a certain plane, so it lies in 2d space. All you need to do is to find a parametrization of the plane in such a way that the use of Pick's theorem on the parametrization gives the required answer.
In other words, find linear function F such that (x,y,z)=F(s,t) and (x0,y0,z0) is a solution to the linear diophantine equation iff (s0,t0) is in Z X Z where (x0,y0,z0)=F(s0,t0)

The fact that F is linear is important since it guarantees the transformation preserves straight lines.