Question 49
Consider two distinct monic cubic polynomials $F(x)=x^3+a_1x^2+a_2x+a_3$ and $G(x)=x^3+b_1x^2+b_2x+b_3$. All real roots of $F(x)=0$, $G(x)=0$, and $F(x)=G(x)$ are written down, giving eight distinct numbers. Prove that the smallest and largest written numbers cannot both be roots of $F$. (I. Bogdanov)