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2002 AMC 12B Problem 1

Problem 1 of 25EasierNumber TheoryArithmetic

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

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Solution

Each of the nine numbers is 10k−1,10^k-1, so their sum is 9+99+⋯+999,999,999.9+99+\cdots+999{,}999{,}999. Dividing by 9,9, M=1+11+111+⋯+111,111,111=123,456,789. \begin{aligned} M &= 1+11+111+\cdots \\ &\quad {}+111{,}111{,}111 \\ &= 123{,}456{,}789. \end{aligned} Its digits are 11 through 9,9, so the missing digit is 0.0. Thus, the correct answer is A.
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Tagged: place value · digits

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