Problem 3. In the surgical department, there are 4 operating rooms: I, II, III, and IV. In the morning, they were all empty. At some point, an operation began in operating room I, then after some time in operating room II, then after some more time in operating room III, and finally in operating room IV.

All four operations ended simultaneously, and the total duration of all operations was 2 hours and 32 minutes. Thirty minutes before the completion of all operations, the total duration of the ongoing operations was 52 minutes, and ten minutes before that, it was 30 minutes. Which operating rooms' operation durations can be determined from these data, and which cannot?
Answer: Only the duration of the operation in Operating Room IV can be determined.

Solution. First, let's prove that the durations of the operations in Operating Rooms I, II, and III cannot be determined uniquely. Indeed, it is not difficult to verify that if the durations of the operations are 70, 39, 33, 10 or 56, 54, 32, 10 minutes, then all conditions of the problem are satisfied. However, in these two variants, the durations of the operations in Operating Rooms I, II, and III are different.

Now let's prove that the duration of the operation in Operating Room IV can be uniquely restored. For this, let's note that the total duration of the operations 40 and 30 minutes before the end of the operations increased by 22 minutes. This means that 30 minutes before the end of the operations, the operations in Operating Rooms I, II, and III were already in progress, otherwise, the total duration would have increased by no more than 20 minutes. Then, by the end of all operations, their total duration is $52 + 30 \cdot 3 = 142$ minutes. Therefore, the operation in Operating Room IV lasted $152 - 142 = 10$ minutes.

Comment. The durations of the operations can be (in minutes) $(70; 36+s; 36-s; 10)$, where $s \in [0; 4]$ or $(55+t; 55-t; 32; 10)$, where $t \in [0; 15]$.