13.115. Tourist $A$ set off from city $M$ to city $N$ at a constant speed of 12 km/h. Tourist $B$, who was in city $N$, upon receiving a signal that $A$ had already traveled 7 km, immediately set off towards him and traveled 0.05 of the total distance between $M$ and $N$ each hour. From the moment $B$ set off until his meeting with $A$, the number of hours that passed is equal to the number of kilometers $B$ traveled per hour. Find the distance between the cities $M$ and $N$, given that it is not less than 100 km.
## Solution.

Let $x$ km be the distance between $M$ and $N$. Then the speed of tourist $B$ is $B-0.05 x$ km/h. Before the meeting, $A$ traveled $7+0.05 x \cdot 12$ km; $B$ traveled $(0.05 x)^{2}$ km. According to the condition, $7+0.05 x \cdot 12+(0.05 x)^{2}=x$, from which $x_{1}=20$ km; $x_{2}=140$ km. Considering that $x \geq 100$ km, we find $x=140$ km.

Answer: $140 \text{ km}$.