
Rectangles
Let be a rectangle and let , ( ) 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,
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 . If you reach the right side, then has integer width (add up all the rightleft travel). Similarily if you reach the top side, then 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: