the cream of the crop of math problems with minimal prereqs

The following description applies to the union of this post and its companion post with physics problems. A few of these are original, many are stolen. There are a few problems here which I independently came up with and later checked that a number of other people had also come up with the problem. The point of the problems is not just to guess the right answer, but to also provide a proof of its correctness (i.e. a compelling argument). A few of the problems are stated a little vaguely, but I think there is a roughly unique way to reasonably resolve the vagueness in each case.

1. You have an \(m\times n\) chocolate tablet. You start breaking it into pieces in some arbitrary order. For instance, you might start by breaking it into an \(m\times 1\) strip and an \(m\times (n-1)\) tablet. You finish when you can’t make any more cuts, i.e. all the pieces are singletons. How many cuts did you make?
2. You have a finite set of points on the plane such that any triangle formed by three of the points has area at most \(1\). Prove that there is a triangle of area \(4\) which contains every point.