IMO Shortlist 1968 problem 25
Dodao/la:
arhiva2. travnja 2012. Let
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
be a real-valued function defined for all real numbers, such that for some
![a>0](/media/m/4/1/b/41bf6a8eeba84545ed84e7cbaea7fbcc.png)
we have
![f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2}](/media/m/c/2/1/c21e772a26af8d6ca07c4a1a84328d79.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
.
Prove that
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
is periodic, and give an example of such a non-constant
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
for
![a=1](/media/m/1/c/6/1c6abdce7cd19174d88d7aa73e680bf7.png)
.
%V0
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have $$f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2}$$ for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Izvor: Međunarodna matematička olimpijada, shortlist 1968