Mock AIME 1 2010 Problems/Problem 5

Problem

For every integer $N$ , the $\emph{balanced ternary}$ representation of $N$ is defined to be the unique sequence of integers $(b_0, b_1, \ldots, b_m)$ , with $b_i \in \{-1, 0, 1\}$ and $b_m \neq 0$ such that $N = \sum_{i=0}^{m} b_i 3^i$ . We represent $N$ as $c_0 c_1 \ldots c_m$ , where $c_i = b_i$ if $b_i$ is 0 or 1, and $c_i = \underline{1}$ if $b_i = -1$ . For example, $2010 = 3^7 - 3^5 + 3^4 - 3^3 + 3^2 + 3 = 10\underline{1}1\underline{1}110$ . Find the last three digits of the sum of all integers $N$ with $1 \leq N \leq 81$ such that $N$ has at least one zero when written in balanced ternary form.

Solution

$\boxed{630}$ .

Mock AIME 1 2010 (Problems, Source)
Preceded by Problem 4	Followed by Problem 6
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Mock AIME 1 2010 Problems/Problem 5

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