Question 1
Let $f(x) = x^{2} + a x + b$ and $g(x) = x^{2} + c x + d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$, and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(2012, -2012)$, what is the value of $a + c$?