1. Sasha went to bed at 10 PM and set the alarm clock (with hands and a 12-hour dial) for 7 AM. During the night, at some point, the alarm clock, which had been working properly, broke, and its hands started moving in the opposite direction (at the same speed). Nevertheless, the alarm rang exactly at the scheduled time in the morning. At what time did the alarm clock break? Justify your answer.

(A. Kuznetsov, D. Shiryayev)
1. Answer: the alarm clock broke at 1 o'clock at night.

Let's imagine that the minute hand on the alarm clock is missing, and the hour hand, at the moment when the alarm clock broke, split into two halves, one of which (as stated in the condition) started moving in the opposite direction, while the other continued its normal movement. Then at 7 o'clock in the morning, both hands will meet! This means that both hands have traveled half the clock face and the split occurred directly "opposite" the 7 mark, i.e., at 1 o'clock at night.