Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.
James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.
a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.
b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.
The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.
The point \(O\) is randomly chosen on piece of square paper. Then the square is folded in such a way that each vertex is overlaid on the point \(O\). The figure shows one of the possible folding schemes. Find the mathematical expectation of the number of sides of the polygon that appears.