Question 114
A set $\mathcal{S}$ consists of $n \ge 1$ positive real numbers, such that for $x \neq y \in \mathcal{S}$:
$$
|y - x| \geq \sqrt{x + y}.
$$
Show that there exists $z \in \mathcal{S}$ such that $z \gt \frac{1}{2}n(n - 1)$.
$$
|y - x| \geq \sqrt{x + y}.
$$
Show that there exists $z \in \mathcal{S}$ such that $z \gt \frac{1}{2}n(n - 1)$.