Question 169
The set $f_1, f_2, \dots, f_n$ of polynomials with real coefficients is called *special* if for all distinct $i, j, k \in \{1, 2, \dots, n\}$ the polynomial $\frac{2}{3}f_i + f_j + f_k$ has no real roots, but for all distinct $p, q, r, s \in \{1, 2, \dots, n\}$ the polynomial $f_p + f_q + f_r + f_s$ has a real root.
1) Give an example of a special set of four polynomials with nonzero sum.
2) Does there exist a special set of five polynomials?
1) Give an example of a special set of four polynomials with nonzero sum.
2) Does there exist a special set of five polynomials?