Rubbanov I.S.

A square is divided by lines into 25 smaller square cells. In some cells, one of the diagonals is drawn such that no two diagonals share a common point (even a common endpoint). What is the maximum possible number of drawn diagonals?

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Example with 16 diagonals, see the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_e8dfada95fe97b3690e8g-10.jpg?height=340&width=332&top_left_y=1691&top_left_x=866)

Estimate. Suppose it was possible to draw 17 diagonals. We will present two ways to arrive at a contradiction.

First method. Each diagonal has two ends located at the nodes of the square grid. There are 36 such nodes in the square. 12 of them are located on the boundary of the inner $3 \times 3$ square (see the left figure), so the number of diagonals with ends in these nodes is no more than 12. The remaining five diagonals can only be located in the central and four corner cells. Therefore, the four nodes located at the vertices of the square are not the ends of the drawn diagonals, which means that 17 diagonals have no more than $36-4=32$ ends. Contradiction.
![](https://cdn.mathpix.com/cropped/2024_05_06_e8dfada95fe97b3690e8g-10.jpg?height=326&width=922&top_left_y=2378&top_left_x=566)

Second method. In each $5 \times 2$ rectangle, no more than 6 diagonals are drawn: there are only six nodes on its middle line, and each diagonal has one of them as its end. Therefore, in all the horizontal rows of the square, except the middle one, no more than 12 diagonals are drawn in total. Therefore, in the middle horizontal row, there are at least 17 - 12 = 5 diagonals, which means that a diagonal is drawn in each of its cells.

Similarly, it can be proven that diagonals are drawn in all cells of the middle vertical row. But diagonals drawn in adjacent (having a common side) cells are parallel. Therefore, the entire "central cross" is filled with parallel diagonals. But then the diagonals in two cells adjacent to the center have a common point (see the right figure). Contradiction.

## Answer

16 diagonals.