# 5. Option 1.

In the district, there are three villages A, B, and C connected by dirt roads, with any two villages being connected by several (more than one) roads. Traffic on the roads is two-way. We will call a path from one village to another either a road connecting them or a chain of two roads passing through a third village. It is known that villages A and B are connected by 34 paths, and villages B and C are connected by 29 paths. What is the smallest number of paths that can connect villages A and C?
Answer: 26.

Solution.

Let there be $k$ roads between cities A and B, $m$ roads between cities B and V, and $n$ roads between cities A and V. Then the number of paths from A to B is $k + m n$, and the number of paths from B to V is $m + k n$. We have the system of equations $k + m n = 34$, $m + k n = 29$, where the unknowns are natural numbers greater than 1. Subtracting the second equation from the first, we get: $(m - k)(n - 1) = 5$. We need to check all divisors of 5: 1 and 5. Thus, $n = 2$ or $n = 6$. For each $n$, we find $k$ and $m$ by solving the original system of linear equations. If $n = 2$, then $k = 8$ and $m = 13$. If $n = 6$, then $k = 4$ and $m = 5$. The number of paths connecting cities A and B is $n + k m$. In the first case, $n + k m = 2 + 8 \cdot 13 = 106$, and in the second case, $n + k m = 6 + 4 \cdot 5 = 26$. Therefore, the desired answer is 26.