7. The greatest integer in a real number x is the integer [x] := n that satisfies n :::; x < n + 1. An interval [a, b] is called Z-asymmetric if b + a -=f. [b] + [a] + 1.

(a) Suppose that R is a two-dimensional Z-asymmetric rectangle; Le., both of its sides are Z-asymmetric. If 'ljJ(x, y) := (x - [x]- 1/2)(y - [y]- 1/2), prove that J J R 'ljJ dA = 0 if and only if R at least one side of R has integer length.

(b) Suppose that R is tiled by rectangles Rl ... , RN , i.e., the Rj's are Z-asymmetric, nonoverlapping, and R = UY=lRj . Prove that if each Rj has at least one side of integer length and R is Z-asymmetric, then R has at least one side of integer length.