OSN 2013 Problem 5

Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that:

• $P(x) = P_1(x) + P_2(x) + P_3(x)$ - $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root)
• The roots of $P_1, P_2, P_3$ are different