Problem 5. Timofey placed 10 grid rectangles on a grid field, with areas of $1, 2, 3, \ldots, 10$ respectively. Some of the rectangles overlapped each other (possibly completely, or only partially). After this, he noticed that there is exactly one cell covered exactly once; there are exactly two cells covered exactly twice; there are exactly three cells covered exactly three times, and exactly four cells covered exactly four times. What is the maximum number of cells that could be covered at least five times? The area of a grid rectangle is the number of cells it contains. Each rectangle lies on the field exactly along the grid cells. (20 points)
Answer: 5 cells.

Solution. Estimation. There are a total of $1+2+\ldots+10=55$ cell coverings. The ten cells described in the condition are covered in total exactly $1 \cdot 1+2 \cdot 2+3 \cdot 3+4 \cdot 4=30$ times. There remain $55-30=25$ cell coverings. Therefore, the number of cells covered at least five times is no more than $25 / 5=5$.

Example. Let's assume that all rectangles have the form $1 \times n$. We will combine them into rectangles of area $15=3+4+8$, $14=1+6+7$, and $12=2+10$. There are still two rectangles of area 5 and 9 left. We will overlay all five rectangles so that their leftmost cells coincide. It is easy to verify that the example works.

Criteria. Example only -10 points.

Estimation only -10 points.