2006 AIME I Problem 6

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

0.\overline{abc} = \frac{100a + 10b + c}{999},

and there are

10 \cdot 9 \cdot 8 = 720

elements to add

By symmetry each digit

0

through

9

appears exactly

72

times in each of the three positions

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.

Problem 6 in other years