6. Find the largest ten-digit number of the form $\overline{a_{9} a_{8} a_{7} a_{6} a_{5} a_{4} a_{3} a_{2} a_{1} a_{0}}$, possessing the following property: the digit equal to $\mathrm{a}_{\mathrm{i}}$ appears in its representation exactly $\mathrm{a}_{9-\mathrm{i}}$ times (for example, the digit equal to $\mathrm{a}_{2}$ appears exactly $\mathrm{a}_{7}$ times).

ANSWER 8888228888.
Solution: We will divide all digits into pairs $a_{0}-a_{9}, \ldots, a_{4}-a_{5}$. We will prove that the digit 9 cannot be present in the given number. Suppose $a_{i}=9$, then the paired digit $a_{9-i}$ appears 9 times. If $a_{9-i} \neq 9$, then 9 appears only once, so we get a number consisting of one nine and nine ones, but then two ones would be in the same pair, which contradicts the condition. If $a_{9-i}=9$, then we get a number consisting of nine nines and one digit not equal to 9, and this digit would form a pair with a nine, which is also impossible.

Let's see if the number can contain 8. If it is paired with 8, then the number contains eight eights, forming pairs, the largest such number is 8888228888. If the paired digit is not 8, then the number contains 8 digits that are less than 8, i.e., it cannot start with four digits 8, so it is less than 8888228888.