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Sum of three square polynomials

Source: Indonesian Mathematical Olympiad 2013 Problem 5

September 5, 2013
algebrapolynomialquadraticsalgebra unsolved

Problem Statement

Let PP be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics P1,P2,P3P_1, P_2, P_3 such that: - P(x)=P1(x)+P2(x)+P3(x)P(x) = P_1(x) + P_2(x) + P_3(x) - P1,P2,P3P_1, P_2, P_3 have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of P1,P2,P3P_1, P_2, P_3 are different
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