Task 3. In the surgical department, there are 4 operating rooms: 1, 2, 3, and 4. In the morning, they were all empty. At some point, a surgery began in operating room 1, after some time - in operating room 2, then after some more time - in operating room 3, and finally in operating room 4.

All four surgeries ended simultaneously, and the total duration of all surgeries was 2 hours and 7 minutes. 18 minutes before the completion of all surgeries, the total duration of the ongoing surgeries was 60 minutes, and 15 minutes before that - 25 minutes. Which operating rooms' surgery durations can be determined from these data, and which cannot?
Answer: Only the duration of the operation in operating room 4 can be determined.

Solution. First, let's prove that the durations of operations in operating rooms 1, 2, and 3 cannot be determined uniquely. Indeed, it is not difficult to verify that if the durations of the operations are 58, 29, 27, 13 minutes or 46, 45, 23, 13 minutes, then all conditions of the problem are satisfied. However, in these two cases, the durations of the operations in operating rooms 1, 2, and 3 are different.

Now let's prove that the duration of the operation in operating room 4 can be uniquely restored. For this, let's note that the total duration of the operations 33 and 18 minutes before the end of the operations increased by 35 minutes. This means that 18 minutes before the end of the operations, the operations in operating rooms 1, 2, and 3 were already in progress, otherwise, the total duration would have increased by no more than 30 minutes. Then, by the end of all operations, their total duration is \(60 + 18 \cdot 3 = 114\) minutes. Therefore, the operation in operating room 4 lasted \(127 - 114 = 13\) minutes.

Comment. The durations of the operations can be \(58; 28+s; 28-s; 13\) minutes, where \(s \in [0; 5]\) or \((45.5+t; 45.5-t; 23; 13)\), where \(t \in [0; 12.5]\).