ILL 1971 Problem 1

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:

(a) rectangles with vertices on the lattice and sides parallel to the coordinate axes;

(b) squares with vertices on the lattice and sides parallel to the coordinate axes;

(c) squares in total, with vertices on the lattice.