There are several squares on a rectangular sheet of chequered paper of size \(m \times n\) cells, the sides of which run along the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square contains another square within itself. What is the largest number of such squares?
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).