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trinomial P_1 + P_2 + P_3 has real roots if every 2 have common root

Source: 2011 Brazil IMO TST 2.1

July 24, 2021
trinomialpolynomialalgebra

Problem Statement

Let P1P_1, P2P_2 and P3P_3 be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial P1+P2+P3P_1 + P_2 + P_3 has also real roots.