Question 100
In a country $2010$ cities are connected directly to the capital by a highway. The number of cities connected directly to any other given city is less than $2010$, and if this number is the same for two cities, then it is even. $k$ of the highways connecting the capital directly to various cities will be closed to traffic for repairs. Find the maximum possible value of $k$ such that this can be done without disrupting the road transportation in the country no matter how the highway network was designed.