[Number Theory]
Problem: Compute the $2011^{\text {th }}$ smallest positive integer $N$ that gains an extra digit when doubled.

Solution: Let $S$ be the set of numbers that gain an extra digit when doubled. First notice that the numbers in $S$ are precisely those whose first digit is at least 5 . Thus there are five one-digit numbers in $S, 50$ two-digit numbers in $S$, and 500 three-digit numbers in $S$. Therefore 5000 is the $556^{\text {th }}$ smallest number in $S$, and because all four-digit numbers greater than 5000 are in $S$, the $2011^{\text {th }}$ smallest number in $S$ is $5000+(2011-556)=\mathbf{6 4 5 5}$.

Final answer: 6455