International Mathematical Olympiad 1968 Problem 5

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $$f(x + a) = \frac{1}{2} + \sqrt{f(x) - [f(x)]^2}$$ holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x + b) = f(x)$ for all $x$).

(b) For $a = 1$, give an example of a non-constant function with the required properties.