Question 49
Let $\mathbb{R}^+$ be the set of positive real numbers. Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function satisfying the following.
For each positive real number $x$, there exists $y \in \mathbb{R}$ such that
$$
(x + f(y))(y + f(x)) \leq 4,
$$
and the number of such $y$'s is finite.
Prove that $f(x) \gt f(y)$ for every pair of positive real numbers $x$ and $y$ with $x \lt y$.
For each positive real number $x$, there exists $y \in \mathbb{R}$ such that
$$
(x + f(y))(y + f(x)) \leq 4,
$$
and the number of such $y$'s is finite.
Prove that $f(x) \gt f(y)$ for every pair of positive real numbers $x$ and $y$ with $x \lt y$.