11. Let's take a positive integer $n$, sum its digits, and then add the digits of this sum again to get an integer $S$. What is the smallest $n$ that allows us to obtain $S \geq 10?$
11. The answer is 199. Indeed, if $n$ had 1 or 2 digits, $n \leq 99$, and then the sum of its digits would be $\leq 18$ and the sum of the digits of the number equal to the sum of its digits would be $\leq 9$, so $n$ has at least three digits. Note that for $n \leq 198$, the sum of the digits of the number is still $\leq 18$. However, if $n=199$, the sum of its digits is 19 and the sum of the digits of 19 is 10, as required.