Question 118
Given Three Quadratic Polynomials $f_1(x) = x^2 + 2a_1x + b_1$, $f_2(x) = x^2 + 2a_2x + b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ such that $a_1a_2a_3 = b_1b_2b_3 \gt 1$. Prove that one of these polynomials has two distinct real roots. Square trinomials $f_1(x) = x^2 + 2a_1x + b_1$, $f_2(x) = x^2 + 2a_2x + b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are given. It is known that $a_1a_2a_3 = b_1b_2b_3 \gt 1$. Prove that at least one of these trinomials has two roots. (N. Agakhanov)