3. In a rectangular grid 303 cells long and 202 cells wide, two diagonals were drawn and all cells through which they passed were painted. How many cells were painted?

(O. A. Pyayve, A. A. Tseler)

![](https://cdn.mathpix.com/cropped/2024_05_06_7886fb0a2000461f3200g-1.jpg?height=123&width=163&top_left_y=2097&top_left_x=1752)
Solution. Mentally divide the large rectangle into $2 \times 3$ rectangles. (The central part of the rectangle is shown in the figure.)

Notice that each diagonal intersects 101 such rectangles (passing through their vertices), and in each of them, it passes through 4 cells. Thus, the two diagonals, it seems, pass through $404 \cdot 2 = 808$ cells. However, the central $2 \times 3$ rectangle is common to both diagonals, and in it, only 6 cells are shaded (not 8 as our calculation suggested).

Answer: 806 cells.

Criteria. The idea of dividing the rectangle into $2 \times 3$ rectangles or indicating that the diagonal passes through their nodes - at least 1 point (but for a "solution" like "in a $2 \times 3$ rectangle, the diagonals intersect 6 cells, so in a $202 \times 303$ rectangle, there will be 101 times more" - 0 points, as there is no idea of dividing into rectangles).

Missing the correction related to the center (i.e., the answer 808) - 3 points; the center is accounted for, but incorrectly - 4 points.

![](https://cdn.mathpix.com/cropped/2024_05_06_7886fb0a2000461f3200g-1.jpg?height=334&width=488&top_left_y=2346&top_left_x=1418)