9. The three-digit number $\overline{a b c}$ consists of three non-zero digits. The sum of the other five three-digit numbers formed by rearranging $a, b, c$ is 2017. Find $\overline{a b c}$.
三位數 $\overline{a b c}$ 由三個非零數字組成。若把 $a 、 b 、 c$ 重新排列, 則其餘五個可組成的三位數之和是 2017 。求 $\overline{a b c}$ 。
9. 425
9. We have $\overline{a b c}+2017=\overline{a b c}+\overline{a c b}+\overline{b a c}+\overline{b c a}+\overline{c a b}+\overline{c b a}=222(a+b+c)$. Since $\overline{a b c}$ is between 111 and 999, 222(a+b+c) is between 2128 and 3016 , so that $a+b+c$ must be 10 , 11,12 or 13 .
If $a+b+c=13$, then $\overline{a b c}=222 \times 13-2017=869$, leading to the contradiction $13=8+6+9$. Similar contradictions arise if $a+b+c$ equals 10 or 12. Finally, $a+b+c=11$, then $\overline{a b c}=222 \times 11-2017=425$ and we check that $4+2+5$ is indeed equal to 11 . Hence the answer is 425 .