IMO Shortlist 1979 problem 8
Dodao/la:
arhiva2. travnja 2012. For all rational
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
satisfying
![0 \leq x < 1](/media/m/e/a/c/eacfaaddba424b9e0cc3ca1704db47f8.png)
, f is defined by
{{ INVALID LATEX }}
Given that
![x = 0.b_1b_2b_3 \cdots](/media/m/a/0/0/a00eac846c40bdc16cec456663ab2e0b.png)
is the binary representation of
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
, find, with proof,
%V0
For all rational $x$ satisfying $0 \leq x < 1$, f is defined by
$$f(x)=\left\{\begin{array}{cc}\frac{f(2x)}{4},&\mbox{ for }0\leq x <\frac{1}{2},\\ \frac{3}{4}+\frac{f(2x-1)}{4}, &\mbox{for }\frac{1}{2}\leq x < 1.\end{array}\right$$
Given that $x = 0.b_1b_2b_3 \cdots$ is the binary representation of $x$, find, with proof, $f(x).$
Izvor: Međunarodna matematička olimpijada, shortlist 1979