Problem - 4221
Select nine different digits from $0$ to $9$ to form a two-digit number, a three-digit number and a four-digit number. The sum of these three numbers is $2017$. Which digit is not selected?
Let the three numbers be $\overline{ab}$, $\overline{cde}$, and $\overline{fghi}$. Then $$a+b+c+\cdots + i\equiv \overline{ab} + \overline{cde}+ \overline{fghi}\equiv 2017\equiv 1\pmod{9}$$
Therefore, if there exists a solution, the missing digit must be $8$. In fact, there exists a solution $$43 + 269 + 1705 = 2017$$
Hence, the answer is $\boxed{8}$.