4. The city has the shape of a grid rectangle $5 \times 10$ cells: lines are streets, cells are residential blocks. The distance between intersections is measured as the length of the shortest path along the city streets, passing from one intersection to another. For example, for intersections $A, B$, and $C$ in the picture, $A B=5, A C=10, B C=9$. Can $k$ intersections be marked in this city so that all distances between these intersections are different numbers? If yes, indicate these intersections (for example, intersection $C$ is at the intersection of the 11th vertical and 6th horizontal streets), if not - explain why it is not possible.

Solve the problem for
a) $k=5$
b) $k=6$
c) $k=7$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7ce0c3c0d23adc3dd673g-1.jpg?height=417&width=759&top_left_y=2264&top_left_x=640)
4. а) It is possible. We can construct the required configuration. It is convenient to choose intersections either only on the very bottom or only on the very right street - then it is easy to calculate distances between them by moving along these streets. In the image below, $1=A B, 2=D E, 3=B C, 4=A C, 9=C D$, $11=C E, 12=B D, 13=A D, 14=B E, 15=A E$.

![](https://cdn.mathpix.com/cropped/2024_05_06_57684bd02359f8f5d463g-1.jpg?height=426&width=797&top_left_y=2123&top_left_x=641)

б) It is possible. Constructing the required configuration is difficult. Here $1=A B, 2=D E, 3=B C, 4=A C$, $5=C F, 6=B F, 7=A F, 8=E F, 9=C D, 10=D F, 11=C E, 12=B D, 13=A D, 14=B E$, $15=A E$.

![](https://cdn.mathpix.com/cropped/2024_05_06_57684bd02359f8f5d463g-2.jpg?height=425&width=799&top_left_y=96&top_left_x=640)

в) It is impossible. In our city, all distances between intersections are whole numbers. The largest of them is the distance between intersections located at opposite corners of the city, which is 15. But seven intersections determine 21 pairwise distances, so there must be identical ones among them.