10. (10 points) Mom decides to take Xiao Hua on a car trip to 10 cities during the holiday. After checking the map, Xiao Hua is surprised to find: among any three of these 10 cities, either all have highways connecting them, or only two cities are not connected by a highway. Therefore, there are at least $\qquad$ highways connecting these 10 cities. (Note: There is at most one highway between any two cities)
【Analysis】Let the 10 cities be: $A 1, A 2, A 3, \cdots, A 10$. First, consider $A 1, A 1$ must have at least 8 highways open, because if $A 1$ only has 7 highways open, then $A 1$ and two of the cities from $A 2 \sim A 10$ will not be connected. Suppose $A 1$ is not connected to $A 2, A 3$, then the three cities $A 1, A 2, A 3$ do not meet the requirement. Similarly, each city must have at least 8 highways open. Therefore, the minimum number of highways between these 10 cities is: $8 \times 10 \div 2=40$.

【Solution】Solution: $8 \times 10 \div 2=40$ (highways)
Answer: Therefore, at least 40 highways are open between these 10 cities.
Hence, the answer is: 40.