Clearly the number of the roads is $3000$. We partition the cities into two groups $A, B$, each group containing exactly $1000$ cities, such that the number of all the roads connecting city in $A$ with city in $B$ is maximal. Suppose there exist two cities $a_1\in A$ and $b_1\in B$, such that $a_1$ has at most one neighbour in $B$ and $b_1$ has at most one neighbour in $A$. Then swapping $a_1$ and $b_1$ we would obtain a configuration with more cross roads that before, which would contradict with the maximal property of the partition $A,B$. Thus, such two cities don't exist. It just means in one of the groups, WLOG say $A$, all the cities have at least two neighbours in $B$. Hence, the number of cross roads is at least $2000$. We choose exactly $2000$ of them and close the other $1000$. Note that all remaining roads connect city in $A$ with city in $B$. In other words we have a bipartite graph, hence there doesn't exist a cycle with odd number of edges (roads).