We have a clock that is uniformly losing time, losing one hour less each day. On the night of January 1, 1982, at 0 o'clock, it showed the exact time, and on the same day, at 13 hours and 5 minutes, the hour and minute hands pointed in the same direction. When will our clock next show the exact time?
Given that the clock is running fast, the synchronization of the two hands does not break, so the minute hand's angular velocity is still twelve times that of the hour hand. This means that the hands overlap at the same angular displacements as in a precise hour, i.e., at integer multiples of the 11th part of the full angle.

According to the condition, our clock runs fast by one hour less per day, so at 13 hours and 5 minutes, it still indicates a time between 13 and 14 hours. Between 13 and 14 hours, the two hands overlap at 13 hours and $60 / 11$ minutes.

Thus, at 13 hours and 5 minutes, our clock indicates 13 hours and $60 / 11$ minutes, meaning it gains $5 / 11$ minutes in 785 minutes. The next time it will show the correct time is when it has indicated exactly 12 hours, or 720 minutes, more than the actual time.

This happens after $785: 5 / 11 \cdot 720 = 1243440$ minutes, or 863.5 days later, which falls on May 13, 1984. (1984 is a leap year.)

Holecz János (Szolnok, Verseghy F. Gymnasium, 2nd grade)