Tokarev S.i.

The path from platform A to platform B was traveled by the electric train in $X$ minutes ($0<X<60$). Find $X$, given that at the moment of departure from $A$ and upon arrival at $B$, the angle between the hour and minute hands was $X$ degrees.
Since more than an hour passes between two consecutive overtakes of the hour hand by the minute hand, no more than one overtake occurred during the specified time of the electric train's movement. Let O be the center of the clock face, and let $T_{A}$ and $T_{B}$ be the points where the tip of the hour hand was located when the electric train passed platforms $A$ and $B$, respectively, and $M_{A}$ and $M_{B}$ be the points where the tip of the minute hand was located at these moments. Note also that in $X$ minutes, the hour hand turns ${ }^{X} / 2$ degrees, and the minute hand turns $6 X$ degrees. Then, if the minute hand overtakes the hour hand, the points $T_{A}, T_{B}, M_{A}$, and $M_{B}$ are arranged relative to each other as shown in the left figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_f3f7c9453eb6f3f37ea7g-33.jpg?height=348&width=590&top_left_y=2338&top_left_x=733)

In this case, the equation $6 X=X+X+X / 2$ must hold, from which $X=0$, which contradicts the problem's condition.

If no overtake occurred, then $T_{A}, T_{B}, M_{A}$, and $M_{B}$ are arranged as shown in the right figure, and the equation

## Answer

$X=48$.