### Instruction: Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \].### Response: Since the problem involves sums of products and squares, I wonder if expanding a binomial square would help.
If I write $x+y+z=0$, then squaring both sides gives $(x+y+z)^2=0$, or $x^2+y^2+z^2+2(xy+yz+zx)=0$.
This means that $xy+yz+zx=-\dfrac{1}{2}(x^2+y^2+z^2)$.