Let $x$, $y$, and $z$ be distinct real numbers that sum to 0. Find the maximum possible value of $$\frac{xy+yz+zx}{x^2+y^2+z^2}.$$
I recall that $(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+zx)$, so if I plug in $x+y+z=0$, I get $x^2+y^2+z^2 = -2(xy+yz+zx)$. This means I can rewrite the fraction as $$\frac{xy+yz+zx}{x^2+y^2+z^2} = -\frac{1}{2}.$$ So the fraction does not depend on the values of $x$, $y$, and $z$, as long as they are distinct and sum to 0. Therefore, the maximum possible value of the fraction is just -1/2.

