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at least one of three euqations has a real root

Source: All-Russian 2007

May 4, 2007
algebrapolynomialquadraticsalgebra proposed

Problem Statement

Given reals numbers aa, bb, cc. Prove that at least one of three equations x2+(a−b)x+(b−c)=0x^{2}+(a-b)x+(b-c)=0, x2+(b−c)x+(c−a)=0x^{2}+(b-c)x+(c-a)=0, x2+(c−a)x+(a−b)=0x^{2}+(c-a)x+(a-b)=0 has a real root. O. Podlipsky
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