The greatest integer in a real number x is the integer [x]: = n which satisfies n < x < n + 1. All interval [a, b] is called Z-asymmetric if b + a ≠ [b] + [a] + 1.
a) Suppose that R is a two-dimensional Z-asymmetric rectangle (i.e., that both of its sides are Z-asymmetric). If ψ(x, y): = (x - [x] - 1/2)(y - [y] - 1/2), prove that ∫∫R ψdA = 0 if and only if at least one side of R has integer length.
b) Suppose that R is tiled by rectangles R1. . . . . .RN (i.e., that the Rj's are Z-asymmetric, nonoverlapping, and that R = UNj=1 Rj). 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.