1. How many times in a day does the angle between the hour and minute hands equal exactly $19^{\circ}$?
Answer: 44.

Solution. Over the time interval from 0:00 to 12:00, the hour hand will make one complete revolution, while the minute hand will make 12 such revolutions. Therefore, during this time, the minute hand will catch up with the hour hand 11 times. Between two consecutive meetings of the hands, the angle between them will be exactly $17^{\circ}$ twice. Thus, between midnight and noon, the required angle between the hands will occur $11 \cdot 2=22$ times, and over a full day, such moments will be twice as many.

Grading. 12 points for a correct solution.