The arithmetic mean of the nine numbers in the set {9, 99, 999, 9999, …, 999999999} is a 9 -digit number M, all of whose digits are distinct. The number M does not contain the digit

2002 AMC 12B Problem 1

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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. The number

M

does not contain the digit

0

2

4

6

8

The mean is

\dfrac{1}{9}(9+99+\cdots+999{,}999{,}999)

Dividing each term by

9

turns the sum into

1+11+111+\cdots

; add these and see which digit never appears

Each of the nine numbers is

10^k-1,

so their sum is

9+99+\cdots+999{,}999{,}999.

Dividing by

9,

\begin{aligned} M &= 1+11+111+\cdots \\ &\quad {}+111{,}111{,}111 \\ &= 123{,}456{,}789. \end{aligned}

Its digits are

1

through

9,

so the missing digit is

0.

Thus, the correct answer is A.