CGMO 2006 Problem 1

Let $a>0$ , the function $f: (0,+\infty) \to R$ satisfies $f(a)=1$ , if for any positive reals $x$ and $y$ , there is $$f(x)f(y)+f \left( \frac{a}{x}\right)f \left( \frac{a}{y}\right) =2f(xy)$$ then prove that $f(x)$ is a constant.