$\large{\begin{array}{ccccccc} && & & & & A&A\\ && & & & & B&B\\ +&& & & & & C&C\\ \hline & & & & & A & B&C\\ \end{array}}$

If each of $A,B,C$ represents a distinct digit from 1 to 9, then what is the value of $\overline{ABC}?$

The answer is 198.

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We are given that $\overline{AA} + \overline{BB} + \overline{CC} = \overline{ABC} \Longrightarrow$

$10A + A + 10B + B + 10C + C = 100A + 10B + C \Longrightarrow B + 10C = 89A$ .

Now since $A,B,C$ are digits the maximum of $B + 10C$ is $99$ , thus we must have $A = 1$ . We then need to find $B,C$ such that $B + 10C = 89$ . Since $B \le 9$ we can only have $C = 8$ , and thus $B = 9$ .

The unique value for $\overline{ABC}$ is therefore $\boxed{198}$ .