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decimal points, periodic fraction representation

Source:

March 5, 2005

Problem Statement

Let MM be the set of the positive rational numbers less than 1, which can be expressed with a 10-distinct digits period in decimal representation. a) Find the arithmetic mean of all the elements in MM; b) Prove that there exists a positive integer nn, 1<n<10101<n<10^{10}, such that n⋅a−an\cdot a - a is a non-negative integer, for all a∈Ma\in M.
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