The problem:

A rectangle R can be tiled with smaller rectangles such that

• The sides of the smaller rectanges are parallel the sides of R.
• At least one side of each of the smaller rectangle is integral.

Show that at least one side of R is integral.

The proof:

Consider $\iint_{R} e^{2 \pi i (x+y)} dxdy$

This is zero, by adding up along each small rectangle. The result follows.