Question 88
Let $p(x) = x^{2013} + a_{2012}x^{2012} + a_{2011}x^{2011} + \dots + a_1x + a_0$ be a polynomial with real coefficients. Suppose that all the roots of $p(x)$ are
$$
-b_{1006}, -b_{1005}, \dots, -b_1, 0, b_1, \dots, b_{1005}, b_{1006}
$$
where $b_1, b_2, \dots, b_{1006}$ are positive real numbers with product equal to $1$. Show that $a_3a_{2011} \geq 1012036$.
$$
-b_{1006}, -b_{1005}, \dots, -b_1, 0, b_1, \dots, b_{1005}, b_{1006}
$$
where $b_1, b_2, \dots, b_{1006}$ are positive real numbers with product equal to $1$. Show that $a_3a_{2011} \geq 1012036$.