$\underline{\text { Folklore }}$

A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus has appeared behind him. As soon as the boy notices the bus, he can change his direction of movement. What is the greatest distance between the stops that guarantees the boy will not miss the bus, if he knows that he is traveling at a speed three times slower than the bus, and he can see the bus at a distance of no more than 2 km?
First solution.

If the boy rides no more than 500 meters away from the bus stop, he will have enough time to return before the bus arrives, which will travel no more than 1500 meters during this time. If the boy rides more than 500 meters away from the bus stop, he will not have enough time to return to it before the bus arrives, so to not miss the bus, he must continue moving to the next stop. If the boy notices the bus when there is no more than 1 km left to the second stop, he will be able to continue moving and arrive at the stop no later than the bus, which will travel no more than 3 km during this time. Thus, the maximum distance between stops, at which the boy is guaranteed not to miss the bus, is 1.5 km.

## Second solution.

If the boy, from the moment he noticed the bus, traveled a distance of \( x \), then the bus traveled a distance of \( 3x \). Suppose the boy rode towards the bus and arrived at the stop simultaneously with the bus, then \( 3x + x = 2 \), hence the boy traveled \( 0.5 \) km to the stop, so if the distance to this stop is more than \( 0.5 \) km, the boy will not make it. If he continued in the same direction he initially traveled and arrived at the stop simultaneously with the bus, then \( 3x - x = 2 \), hence the boy traveled 1 km, so if the distance to this stop is more than 1 km, the boy will not make it. Thus, the distance between stops should not exceed \( 1.5 \) km.

## Answer

1.5 km.