Solution:

Assume otherwise for the sake of contradiction, i.e. that there exist $x, y, z \in S$ so that $x + y \leq z$. For fixed $z$, the product $x y$ is then maximized when $x = y$. Then, if $x, y, z$ have average $a$, their product is at most that when $x + y = z$, which happens at $x = 3a/4$, $y = 3a/4$, $z = 3a/2$, for a product of $27a^3/32$.

Now consider replacing all of $x, y, z$ with $a$ in $S$. The product multiplies by at least $32/27$ for a product of at least $0.9 \cdot 32/27 > 1$, while the average is unchanged, a contradiction to AM-GM.