11. For every integer $n$, let $S(n)$ be the sum of the digits of $n$ (in base 10). What is the smallest integer $N$ for which $S(S(N)) \geq 10$?
11. The answer is 199. Indeed, if $n$ had 1 or 2 digits, $n \leq 99$, and then $S(n) \leq 18$ and $S(S(n)) \leq 9$, so $n$ must have at least three digits. We observe that for $n \leq 198$, still $S(n) \leq 18$. However, if $n=199, S(n)=19$ and $S(S(n))=10$, as required.