2006 IMO Shortlist Problems/N2
Problem
(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
.
Solution
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
,
,
and
,
so
,
and
.
Hence is rational. ∎
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