3. (7 points) At a certain moment, Anya measured the angle between the hour and minute hands of her clock. Exactly one hour later, she measured the angle between the hands again. The angle turned out to be the same. What could this angle be? (Consider all cases.)
Answer: $15^{\circ}$ or $165^{\circ}$.

Solution. After 1 hour, the minute hand remains in its original position. During this time, the hour hand has turned $30^{\circ}$. Since the angle has not changed, the minute hand must bisect one of the angles between the positions of the hour hand (either the $30^{\circ}$ angle or the supplementary angle of $330^{\circ}$).
![](https://cdn.mathpix.com/cropped/2024_05_06_66cba580fc08050db9edg-2.jpg?height=452&width=1052&top_left_y=131&top_left_x=570)

Therefore, the hour hand was either at $15^{\circ}$ earlier or at $165^{\circ}$ later.

Criteria. Any correct solution: 7 points.

Both correct answers given without justification or with incorrect justification: 3 points.

One of the correct answers given: 1 point.