The Tarski Plank Problem:

Unit disc can be covered by $n$-rectangles $1\times n^{-1}$. Prove that it cannot be covered by a smaller number of such rectangles.

Solutions. Take the unit sphere of the same center as the disc. Let $\pi$ be the orthogonal projection on the plane containing the disc. Taking $\pi^{-1}$ of a strip (assuming not too much of it is "outside") we get a "ring" on the sphere. The point is that the area of that ring depends only on its width which is $n^{-1}$ and not on its position. This is the Archimedes Hat-Box Theorem. Since we cover sphere by sets of equal areas, they cannot overlap and the problem follows. $\Box$

Of course, this solution is too short, only an idea.