ABC

From the equation, we know that A+B+C=C+10x and x is an unknown positive integer. Because both A and B are non-zero so x≠0, and the max sum of A and B is 17, so x≤1.7, which means the only value of x is 1(x=1). And then we know that A+B=10x=10. so $\overline{AA}$ $+$ $\overline{BB}$ $=110$ . Because of $\overline{CC}$ must be no larger than 99. As a result, A can only be 1, and B can only be 9. Now $\overline{ABC}$ can be written as $\overline{19C}$ because $\overline{AA}$ $\overline{BB}$ and $\overline{CC}$ are all mutilple of 11, and 198 is larger but the closest one to $\overline{19C}$ .So we got C=8

Now is the moment of truth: $\overline{AA}$ $+$ $\overline{BB}$ $+$ $\overline{CC}$ $=11$ $+$ $99$ $+$ $88$ $=198$