Question 43
Consider nine monic quadratic polynomials $x^2+a_ix+b_i$, $i=1,\ldots,9$. Both $(a_i)$ and $(b_i)$ are arithmetic progressions, and the sum of the nine polynomials has at least one real root. Let $k$ count the original polynomials having no real roots. Find the maximum possible $k$. (I. Bogdanov)