1

Part of 1987 Balkan MO

Problems(1)

functional equation f(x+y)=f(x)f(a-y)+f(y)f(a-x) - show f is constant

Source: bmo 1987

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