2006 IMO Shortlist Problems/N2
(Canada) For let be the number whose th digit after the decimal point is the th digit after the decimal point of . Show that if is rational then so is .
For any real and any natural number , let the th digit after the decimal point of . We note that is rational if and only if is periodic for sufficiently large , i.e., if is determined by the residue of mod , for some integer .
Suppose is rational, and let be an integer such that for sufficiently large , is determined by the residue of mod . Let , for some odd integer and some nonnegative integer . We note that the residue of mod is uniquely determined by the residues of mod and mod . Then for sufficiently large ,
Hence is rational. ∎
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