Question 84
Determine all functions $f:(0,+\infty) \rightarrow \mathbb{R}$ such that $f(2009)=1$ and
$$
f(x) f(y)+f\left(\frac{2009}{x}\right) f\left(\frac{2009}{y}\right)=2 f(x y)
$$
for all positive real numbers $x$ and $y$.
$$
f(x) f(y)+f\left(\frac{2009}{x}\right) f\left(\frac{2009}{y}\right)=2 f(x y)
$$
for all positive real numbers $x$ and $y$.