Harvard-MIT Mathematics Tournament

Solution:
Write the elements of $S$ in their ternary expansion (i.e. base $3$). Then the second condition translates into: if $\overline{d_{1} d_{2} \cdots d_{k}} \in S$, then $\overline{d_{1} d_{2} \cdots d_{k} 0}$ and $\overline{d_{1} d_{2} \cdots d_{k} 1}$ are also in $S$. It follows that $S$ is the set of nonnegative integers whose ternary representation contains only the digits $0$ and $1$.

Since $2 \cdot 3^{6} < 2008 < 3^{7}$, there are $2^{7} = 128$ such elements less than $2008$. Therefore, there are $128$ such non-negative elements.