Problem 1. Two positive numbers $x$ and $y$ are written on the board in non-decreasing order, i.e., $x \leq y$. Maria writes $x^{2}$ in her notebook and then replaces the numbers on the board with $x$ and $y-x$, written again in non-decreasing order. She repeats the same operation with the new numbers and so on until one of the two numbers on the board becomes 0. What will be the sum of the numbers in Maria's notebook?
Solution: Let's consider a rectangle with sides $x$ and $y$. At the first step, we cut off a square with side $x$ from the rectangle and record its area, at the second step we perform the same operation with the new rectangle, and so on. Eventually, the rectangle will be cut into squares, whose areas are recorded. The sum of these numbers is equal to the area of the given rectangle, i.e., $x y$.