IMO Shortlist 2006 problem N2
Dodao/la:
arhiva2. travnja 2012. For
![x \in (0, 1)](/media/m/0/5/2/052d82525657e07cf26dc1fc09f7e360.png)
let
![y \in (0, 1)](/media/m/d/c/9/dc93edaeaf7b5a3406d1a4e29b4b1c5f.png)
be the number whose
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
-th digit after the decimal point is the
![2^{n}](/media/m/e/1/c/e1c9f62ce706ce3101a31898470616c3.png)
-th digit after the decimal point of
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
. Show that if
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
is rational then so is
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
.
%V0
For $x \in (0, 1)$ let $y \in (0, 1)$ be the number whose $n$-th digit after the decimal point is the $2^{n}$-th digit after the decimal point of $x$. Show that if $x$ is rational then so is $y$.
Izvor: Međunarodna matematička olimpijada, shortlist 2006