
Rectangles
Let $\displaystyle R = [a,b]\times[c,d]$ be a rectangle $\displaystyle ( d(ab) = d(cd); d(ac) = d(bd) )$ and let $\displaystyle R_i = (a_i,b_i)\times(c_i,d_i)$, ($\displaystyle 1\leq i \leq n$) be rectangles inside R, such that every two rectangles in R are disjoint, maybe other than their sides, and such that each one of them has at least one integer side.
Also,
$\displaystyle \bigcup_{1\leq i \leq n}R_i = R$
Prove that R also has at least one integer side.

We'll consider the whole thing as a possibly quite irregular grid on which we will move. A "point" will mean the vertex of some rectangle, and "moving up once" will mean to move up from a point to the next point directly above, along a side of integer length.
First notice that if a rectangle has one side of integer length then the opposite side is also of integer length. Therefore, given two concurrent sides of a rectangle, one will be of integer length.
Start from the lower left corner. From there you can certainly move up or move right; one of those sides has to be of integer length. From the point you reach then, the same is possible : you can move either up or right. Keep doing that until you reach either the right side or the top side of $\displaystyle R$. If you reach the right side, then $\displaystyle R$ has integer width (add up all the rightleft travel). Similarily if you reach the top side, then $\displaystyle R$ has integer height. Since one of the two must happen, we are done.

Very nice! I haven't quite thought about it this way. There are numerous other ways of solving this, so anyone is welcome to have a shot!
Another nice solution is by using the fact that:
$\displaystyle \sum^{n}_{i=1} \int_{R_i} sin(2\pi x)sin(2\pi y)dxdy = \int_{R}sin(2\pi x)sin(2\pi y) dxdy$