2002 AMC 10B Problem 3

Problem 3 of 25EasierAlgebra Number Theory

The arithmetic mean of the nine numbers in the set

\{9, 99, 999, 9999, \ldots, 999999999\}

is a

9

-digit number

M,

all of whose digits are distinct. Which digit does the number

M

not contain?

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Solution

The mean is

\begin{aligned} &\dfrac{1}{9} \\ &\quad {}\cdot \small \left(9 + 99 + 999 + \cdots + 999999999\right) \\ &= 1 + 11 + 111 + \cdots \\ &\quad {}+ 111111111. \end{aligned}

Adding these nine repunits column by column gives

M = 123456789.

The only digit missing from

M

0.

Thus, the correct answer is A.