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 — Solution

2002 AMC 12B Problem 1

Below is the professionally curated solution for Problem 1 of the 2002 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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Concepts:place value digits

Difficulty rating: 950

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

$0$

$2$

$4$

$6$

$8$

Solution:

Each of the nine numbers is $10^k-1,$ so their sum is $9+99+\cdots+999{,}999{,}999.$ Dividing by $9,$ $M=1+11+111+\cdots+111{,}111{,}111=123{,}456{,}789.$ Its digits are $1$ through $9,$ so the missing digit is $0.$

Thus, the correct answer is A.

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