1. In a $5 \times 5$ square, color as many cells black as possible, so that the following condition is met: any segment connecting two black cells must necessarily pass through a white cell.
Answer. We will prove that it is impossible to color more than 9 cells. Divide the square into 9 parts:

![](https://cdn.mathpix.com/cropped/2024_05_06_7976d8a667797e2cc004g-3.jpg?height=289&width=262&top_left_y=181&top_left_x=974)

In each part, no more than one cell can be colored.

9 cells can be colored as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_7976d8a667797e2cc004g-3.jpg?height=380&width=388&top_left_y=598&top_left_x=911)

Comment. A correct solution, including justification that the number of colored cells is maximal, and a diagram - 20 points. An incomplete justification that the number of colored cells is maximal - 15 points. Only an answer in the form of a diagram - 10 points. The solution is started but progress is insignificant - 1 point.

2. A baby squirrel had a reserve of pine and spruce cones, with pine cones making up 60%. In August, it collected more spruce cones, and now the pine cones made up 20%. In September, the baby squirrel collected more pine cones, and now the pine cones made up 80%. By what factor did the total number of cones increase over the two months?

## Answer. 12.

Solution. In August, the percentage of pine cones decreased by 3 times, meaning the total number of cones increased by 3 times. In September, the percentage of spruce cones decreased from 80% to 20%, meaning the total number of cones increased by 4 times. In total, over the two months, the total number of cones increased by \(3 \cdot 4 = 12\) times.

Comment. A correct solution - 20 points. There is a gap in the justification - 15 points. A correct equation is set up, but not brought to a correct solution - 10 points. The solution is started, with some progress - 5 points. The solution is started, but progress is insignificant - 1 point.