Question 12
Let $P(x) = x^{2012} + a_{2011}x^{2011} + a_{2010}x^{2010} + \dots + a_1x + a_0$ be a polynomial of degree $2012$ of real coefficients with $1$ as its leading coefficient. Find the minimum of real number $c$ such that $|\operatorname{Im} z| \le c |\operatorname{Re} z|$, where $\operatorname{Re} z$ and $\operatorname{Im} z$ are, respectively, the real and the imaginary parts of any root of a polynomial obtained by changing some of the coefficients of $P(x)$ to their opposite numbers. (posed by Zhu Huawei)