7.5. In the city, there are 9 bus stops and several buses. Any two buses have no more than one common stop. Each bus has exactly three stops. What is the maximum number of buses that can be in the city
Answer: 12.

Solution: if some stop is common for 5 routes, then no two of them have any more common stops, which means there are at least $1+5*2=11$ stops, which contradicts the condition. Therefore, each stop is the intersection of no more than 4 routes, then the total number of stops made by buses is no more than 94, and since there are 3 buses on each route, the total number of buses is no more than $94 / 3=12$.

Buses in the city can run, for example, as follows (stops are marked with numbers): $(1,2,3),(4,5,6),(7,8,9)$ $(1,4,7),(2,5,8),(3,6,9)$ $(1,5,9),(2,6,7),(3,4,8)$ $(3,5,7),(2,4,9),(1,6,8)$.

Criteria: only the example - 3 points; only the estimation that it cannot be more than 12 - 3 points; only the remark that one stop cannot be in more than 4 routes - 1 point; only the answer - 0 points.

## 8th grade

## Each problem is worth 7 points