BALKAN 1987 Problem 1

Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and $$f(x+y)=f(x)f(a-y)+f(y)f(a-x), \forall x,y \in \mathbb{R}.$$ Prove that $f$ is constant.