1. The Niva and Toyota cars are driving on a circular test track, a quarter of which is a dirt road, and the remaining part is paved. The Niva's speed on the dirt road is 80 km/h, and on the paved road - 90 km/h. The Toyota's speed on the dirt road is 40 km/h, and on the paved road - 120 km/h. The cars start simultaneously at the beginning of the dirt section and first drive on this dirt section. On which lap will one of the cars overtake the other for the first time?
Answer: The Niva will overtake the Toyota on its $11-$th lap while the Toyota is on its $10-$th lap. Solution. Let $S$ km be the length of the dirt road section, then $3 S$ km is the length of the asphalt section. The Niva completes one lap in $\frac{S}{80}+\frac{3 S}{90}=\frac{11 S}{240}$ hours, while the Toyota completes one lap in $\frac{S}{40}+\frac{3 S}{120}=\frac{12 S}{240}>\frac{11 S}{240}$ hours. Since the cars start on the dirt road where the Niva's speed is higher, the Niva will initially take the lead and eventually overtake the Toyota, and the overtaking can only occur on the dirt road where the Niva's speed is higher.

Let's assume that by the time of the first overtaking, the Toyota will have completed $n$ full laps and an additional distance of $S \cdot x$, where $0<x<1$ (if $x=1$, no overtaking will occur: the cars will be side by side, but after entering the asphalt road, the Toyota will be faster than the Niva). Then, by the time of the overtaking, the time the Toyota has been traveling will be $\frac{12 S}{240} \cdot n+\frac{S \cdot x}{40}$, and the time the Niva has been traveling will be $\frac{11 S}{240} \cdot(n+1)+\frac{S \cdot x}{80}$. Since the cars start simultaneously, we get the equation $\frac{12 S}{240} \cdot n+\frac{S \cdot x}{40}=\frac{11 S}{240} \cdot(n+1)+\frac{S \cdot x}{80}$, from which $12 n+6 x=11 n+11+3 x$, or $n+3 x=11$. Since $0<x<1$, it follows that $n=9$ or $n=10$. Therefore, by the time of the first overtaking, the Toyota will have completed 9 full laps and will be transitioning to the $10-$th, while the Niva will have completed 10 full laps and will be transitioning to the $11-$th.