IMO 1968 Problem 5

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 $a=1$ .