KZ-MO 2006 Problem 2

Product of square trinomials $x ^ 2 + a_1x + b_1$ , $x ^ 2 + a_2x + b_2$ , $\dots$ , $x ^ 2 + a_n x + b_n$ equals polynomial $P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n}$ , where the coefficients $c_1$ , $c_2$ , $\dots$ , $c_ {2n}$ are positive. Prove that for some $k$ ( $1 \leq k \leq n$ ) the coefficients $a_k$ and $b_k$ are positive.