Question 82
Let $\mathbb{R}_{\gt 0}$ be the set of positive real numbers. Let $a \in \mathbb{R}_{\gt 0}$ be given. Find all functions $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}$ such that $f(a)=1$ and
$$
\forall x, y \in \mathbb{R}_{\gt 0}: f(x) f(y)+f\left(\frac{a}{x}\right) f\left(\frac{a}{y}\right)=2 f(x y)
$$
$$
\forall x, y \in \mathbb{R}_{\gt 0}: f(x) f(y)+f\left(\frac{a}{x}\right) f\left(\frac{a}{y}\right)=2 f(x y)
$$