Problem 1. In the country of "Energetika," there are 150 factories, and some of them are connected by bus routes that do not stop anywhere except at these factories. It turned out that any four factories can be divided into two pairs such that there is a bus route between the factories of each pair. Find the smallest number of pairs of factories that can be connected by bus routes.
Solution. Suppose that some factory $X$ is connected by bus routes to no more than 146 factories. Then a quartet of factories, consisting of $X$ and any three with which it is not connected, does not satisfy the problem's condition, since $X$ cannot be paired with any of the three remaining factories. Therefore, each factory is connected to at least 147 factories. Consequently, the total number of pairs of factories connected by bus routes is no less than $\frac{147 \cdot 150}{2}=11025$.

Now, we will show that there can be exactly 11025 pairs of factories. Number the factories from 1 to 150 and connect by bus routes all factories except the first and the 150th, as well as factories whose numbers differ by one. We will check that this construction satisfies the problem's condition. Since each factory is connected by bus routes to 147 factories, the total number of connected factory pairs is exactly $\frac{147 \cdot 150}{2}=11025$.

Now, take any quartet of factories. There are two cases.

1) There is a factory not connected to two of the other three factories. Let factory A not be connected to factories B and C, but connected to factory D. Then factories B and C must be connected to each other, since the remainders of their numbers when divided by 150 differ by 2. Therefore, the pairs (A,D) and (B,C) are suitable.
2) All factories are connected to at least two of the other three factories. Let factory A be connected to factories B and C. By assumption, factory D must be connected to B or C. If it is connected to B, then the pairs (A,C) and (B,D) are suitable, and if it is connected to C, then the pairs (A,B) and (C,D) are suitable.

Answer: 11025.