2022 AMC 10A Problems/Problem 17

Problem

How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy $0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$ (The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$

$\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$

Solution

Notice that $0.\overline{abc} = \frac{abc}{999}$ and $0.\overline{x} = \frac{x}{9}$ . From this, we can write:

Solution in progress

~KingRavi

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2022 AMC 10A Problems/Problem 17

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