A large rectangle is partitioned into smaller rectangles, with sides parallel to those of the large rectangle. Each smaller rectangle has at least one side whose length is an integer. (For some rectangles this may be the horizontal side; for others, the vertical side.)
Prove that the same holds true of the large rectangle.
Update (09/03/2008): Solution posted! (show solution)
Consider the following integral:
$$\int_{x_0}^{x_1}\int_{y_0}^{y_1}e^{2\pi i(x+y)}dxdy = $$
$$\int_{x_0}^{x_1}e^{2\pi ix}dx \int_{y_0}^{y_1}e^{2\pi iy}dy = $$
$$\frac{(e^{2\pi ix_1} - e^{2\pi ix_0}) (e^{2\pi iy_1} - e^{2\pi iy_0})}{(2\pi i)^2} $$
It is easy to see that this is $0$ if and only if $x_1-x_0$ or $y_1-y_0$ is an integer. The value of the integral for each small rectangle is $0$, so it has to be $0$ also for the big rectangle, that must now have at least one side of integer lenght.
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