2. (6 points) It is known that no digit of a three-digit number is zero and the sum of all possible two-digit numbers formed from the digits of this number is equal to this number. Find the largest such three-digit number.
# Answer: 396 

Solution. If there are identical digits, then the sum of three two-digit numbers is less than 300. If the digits are different, then $\overline{a b c}=\overline{a b}+\overline{a c}+\overline{b a}+\overline{b c}+\overline{c a}+\overline{c b}=22(a+b+c)$. The sum of the digits gives the same remainder when divided by 9 as the number itself. Therefore, $a+b+c$ is divisible by 9, and thus does not exceed 18. Then $\overline{a b c} \leq 22 \cdot 18=396$ fits.