25. In 80 cities, there are two types of flight routes being executed: any city is directly connected to at least 7 other cities; any two cities can be connected through a finite number of direct flights. Find the smallest positive integer $k$, such that no matter how the routes satisfying the conditions are arranged, any city can reach any other city with at most $k$ direct flights.
25. Suppose there are two cities $A_{1}$ and $A_{k+1}$ that require at least $k$ stops to reach each other. Let the shortest route from $A_{1}$ to $A_{k+1}$ be $A_{1} \rightarrow A_{2} \rightarrow A_{3} \rightarrow \cdots \rightarrow A_{k} \rightarrow A_{k+1}$. Since each city is directly connected to at least 7 other cities, $A_{1}$ and $A_{k+1}$, in addition to the cities in the set $\{A_{1}, A_{2}, \cdots, A_{k+1}\}$, are each directly connected to at least 6 other cities. Each city in $A_{2} \sim A_{k}$, besides the cities in the set $A$, is directly connected to at least 5 other cities. Let the sets of cities directly connected to $A_{1}, A_{4}, A_{7}, A_{10}, \cdots, A_{3m-2}, A_{k+1}$ (where $3m+1 \leq k \leq 3m+2$) and not in $A$ be denoted as $X_{i} (i=0,1,2, \cdots, m)$. It is easy to see that $\left|X_{0}\right| \geq 6, \left|X_{m}\right| \geq 6, \left|X_{i}\right| \geq 5 (1 \leq i \leq m-1)$. Also, $\left|X_{i} \cap X_{j}\right| = \varnothing (0 \leq i < j \leq m)$, otherwise there would be a shorter route between $A_{1}$ and $A_{k+1}$. Therefore, $\left|A \cup (X_{0} \cup X_{1} \cup \cdots \cup X_{m})\right| \geq k+1 + 6 \times 2 + (m-1) \times 5 \geq 8m + 9$. If $8m + 9 \geq 81$, i.e., $m \geq 9, k \geq 28$. This implies there must be at least 81 cities, which is a contradiction. Hence, $k \leq 27$. 

Next, when $k=27$, consider 28 cities $A_{1}, A_{2}, \cdots, A_{28}$ and the city sets $X_{i} (i=0,1, \cdots, 9)$, where $\left|X_{0}\right| = \left|X_{9}\right| = 6, \left|X_{i}\right| = 5 (1 \leq i \leq 8)$, and for $0 \leq i < j \leq 9$, $X_{i} \cap X_{j} = \varnothing$, and $X_{j}$ does not include the cities $A_{1}, A_{2}, \cdots, A_{28}$. In this case, there are a total of $28 + 2 \times 6 + 8 \times 5 = 80$ cities. For $1 \leq k \leq 8$, $A_{3k}, A_{3k+1}, A_{3k+2}$ are directly connected to all cities in $X_{k}$; $A_{1}, A_{2}$ are directly connected to all cities in $X_{0}$; $A_{27}$ and $A_{28}$ are directly connected to all cities in $X_{9}$. Any city in $X_{i} (0 \leq i \leq 9)$, besides being directly connected to the aforementioned $A_{t}$ cities, is only directly connected to the other cities in $X_{i}$. $A_{i}$ is directly connected to $A_{i+1} (i=1,2, \cdots, 27)$. This way, each city in $A_{1}$ to $A_{28}$ is directly connected to at least 7 other cities, and any city in $X_{i}$ is directly connected to exactly 7 other cities, and it takes at least 27 direct flights to connect from $A_{1}$ to $A_{28}$. Therefore, the smallest positive integer $k$ is 27.