# 2013 UNCO Math Contest II Problems/Problem 9

## Problem

The standard abbreviation for the non-terminating repeating decimal $.34121121121121121\ldots$ is $.34\overline{121}$, a string of five digits. How many distinct non-terminating repeating decimals $.d_1d_2d_3\ldots$ have standard abbreviations that have at most six digits? (Consider two nonterminating decimals distinct if they differ in any digit. Nonterminating means that the digits are not eventually all zero.) COMMENTS The standard abbreviation is also the shortest. For example, $.34121121121121121\ldots= .34\overline{121}$ can also be abbreviated as $.341\overline{211}$, or as $.3412\overline{112}$, or as $.34121\overline{121}$ by sliding the bar rightward, making longer strings. The nonterminating decimal $.34\overline{121}$ has two parts: a repeating tail $T = \overline{121}$ and a non-repeating head $H = 34$. If the string has no head, the decimal is periodic, which is acceptable. There must be a tail string $T$, which by convention is NOT permitted to be $T = \overline{0}$, since that corresponds to a terminating decimal. The examples $.\overline{345}$, $.\overline{9}$, and $.721\overline{9}$ are all standard abbreviations for nonterminating repeating decimals.

## Solution $5,778,810$