Ukrainian National Mathematical Olympiad

It is almost obvious that both numbers should be five-digit, since otherwise their difference will have at least five digits. Indeed, the smallest six-digit number that we can build is $102345$, the biggest four-digit number is $9876$, and their difference is $92469$. For other cases of integers with different number of digits the difference will be even bigger than that.

In order to minimize the difference of two five-digit numbers we should take their first digits to be consequent, because otherwise their difference will also have five digits. Now we need to consider all possibilities when a number with greater first digit is complemented by the minimal digits possible, and the other number is complemented by the maximal digits possible. $$\begin{gathered} 90123-87654=2469,80123-79654=469,70123-69854=269,60123-59874=249, \\ 50123-49876=247,40125-39876=249,30145-29876=269,20345-19876=469. \end{gathered}$$ We see that the minimal possible difference is attained for the numbers $50123$ and $49876$, and is equal to $247$.