Problem 3. In a surgical department, there are 4 operating rooms: A, B, V, and G. In the morning, they were all empty. At some point, an operation began in operating room A, after some time - in operating room B, then after some more time - in V, and then in $\Gamma$.

All four operations ended simultaneously, and the total duration of all operations was 3 hours and 5 minutes. 36 minutes before the completion of all operations, the total duration of the ongoing operations was 46 minutes, and 10 minutes before that - 19 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 $\Gamma$ can be determined.

Solution. First, let's prove that the durations of operations in operating rooms A, B, and C cannot be determined uniquely. Indeed, it is easy to verify that if the durations of the operations are $65, 45, 44, 31$ or $56, 55, 43, 31$ minutes, then all conditions of the problem are satisfied. However, in these two cases, the durations of the operations in operating rooms A, B, and C are different.

Now let's prove that the duration of the operation in operating room Г can be uniquely determined. For this, let's note that the total duration of the operations 46 and 36 minutes before the end of the operations increased by 27 minutes. This means that 36 minutes before the end of the operations, the operations in operating rooms A, B, and C 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 $46 + 36 \cdot 3 = 154$ minutes. Therefore, the operation in operating room Г lasted $185 - 154 = 31$ minutes.

Comment. The durations of the operations can be ( $65 ; 44.5+s ; 44.5-s ; 31$ ), where $s \in [0 ; 1.5]$ or ( $55.5+t ; 55.5-t ; 43 ; 31$ ), where $t \in [0 ; 9.5]$.