Question

Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a,$ $b,$ $c$ are distinct digits. Find the sum of the elements of $\mathcal{S}.$

Answer

Answer: 360

Concepts:repeating decimal digits symmetry

Difficulty rating: 2300

Solution:

Each element equals $0.\overline{abc} = \frac{100a + 10b + c}{999},$ and there are $10 \cdot 9 \cdot 8 = 720$ ordered triples of distinct digits. By symmetry, each digit $0$ through $9$ appears in each of the three positions exactly $\frac{720}{10} = 72$ times.

The numerators therefore total $\begin{aligned} &72 (0 + 1 + \cdots + 9) \\ &\quad {}\cdot (100 + 10 + 1) \\ &= 72 \cdot 45 \cdot 111 = 359640, \end{aligned}$ so the sum of the elements is $\frac{359640}{999} = 360.$

2006 AIME I Problem 6

Answer: 360

Difficulty rating: 2300

Solution:

Problem 6 in Other Years