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Three quadratic polynomials

Source: Problem 1 of Russian Regional Olympiad 2010 Grade 9

September 25, 2011
quadraticsalgebrapolynomialalgebra proposed

Problem Statement

Three quadratic polynomials f1(x)=x2+2a1x+b1f_1(x) = x^2+2a_1x+b_1, f2(x)=x2+2a2x+b2f_2(x) = x^2+2a_2x+b_2, f3(x)=x2+2a3x+b3f_3(x) = x^2 + 2a_3x + b_3 are such that a1a2a3=b1b2b3>1a_1a_2a_3 = b_1b_2b_3 > 1. Prove that at least one polynomial has two distinct roots.
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