The way to have one digit appear twice as many times as the other nine is to set a period of 11 digits. To do that, we need the fraction to be X/99999999999. Factoring the denominator leads us to 21649 as the denominator, and looking at the possible numerators we find that 639/21649 solves the first two conditions.
To get fewer number of distinct digits, 646/21649 may seem to be the best solution (there are only two different digits), but using improper fractions, we can get the best solution: 333333/21649.
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