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Two monic trinomials

Source: All-Russian MO 2001 Grade 11 #5

January 3, 2012
quadraticsalgebrapolynomialalgebra unsolved

Problem Statement

Two monic quadratic trinomials f(x)f(x) and g(x)g(x) take negative values on disjoint intervals. Prove that there exist positive numbers α\alpha and β\beta such that αf(x)+βg(x)>0\alpha f(x) + \beta g(x) > 0 for all real xx.
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