I.V. Yakheneno

Petya and Vitya were going down the escalator. In the middle of the escalator, the hooligan Vitya snatched Petya's hat and threw it onto the opposite escalator. The victim Petya ran back up the escalator to then go down and retrieve the hat. The cunning Vitya ran down the escalator to then go up and get there before Petya. Who will get there first, if the speeds of the boys relative to the escalator are constant and do not depend on the direction of movement?
Two opposite escalators can be imagined as a constantly moving ring, relative to which the hat is stationary. Look at what is happening from the point of view of the hat.

## Solution

Two opposite escalators actually form a constantly moving ring (on which you can ride like a carousel), relative to which the hat is stationary. Stand next to the hat and observe the boys' run. In this case, we can assume that the escalators are stationary, and the boys are running towards us from the diametrically opposite point of the ring with equal speeds, but each from their own side. Now it is obvious that they will reach the hat simultaneously.

However, these considerations are valid under one condition: Petya must reach the top of the escalator before the hat gets there (the hat itself cannot switch to Petya's escalator and ride towards him).

## Answer

| If the boys' speeds are at least twice the speed of the escalator, they will reach the hat simultaneously. Otherwise, Vitya will reach the hat first.