Silk Road Mathematics Competition

1) The polynomials $x^2+x+100$, $x^2+x+100$, $-x^2+x-200$, $1$ give a needed example.

2) To the contrary, assume that there exists a special set of polynomials $f_1,f_2,f_3,f_4,f_5$ with real coefficients. Consider a complete graph with a set of vertices $\{1,2,3,4\}$. An edge $(i,j)$ is called positive (negative), if the polynomial $\frac{2}{3}{f}_5+f_i+f_j$ is positive (negative). By the assumption in this graph there is no any positive (negative) triangles. Indeed, if for example $\frac{2}{3}{f}_5+f+g>0$, $\frac{2}{3}{f}_5+g+h>0$ and $\frac{2}{3}{f}_5+f+h>0$, then $f_5+f+g+h>0$ which is a contradiction with the definition of a special set. The next lemma can be easily proved.

Lemma. In a two-colored complete graph with four vertices without one-colored triangles there is a pair of skew (crossed) edges of every color.

Using the lemma we have conflicting inequalities: $$f_1+f_2+f_3+f_4+\frac{4}{3}{f}_5>0,$$ $$f_1+f_2+f_3+f_4+\frac{4}{3}{f}_5<0.$$