3. Let $x, y$, and $z$ be distinct real numbers that sum to 0 . Find the maximum possible value of
$$
\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}} \text {. }
$$
Solution:
$$
-1 / 2
$$

Note that $0=(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$. Rearranging, we get that $x y+y z+z x=-\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right)$, so that in fact the quantity is always equal to $-1 / 2$.