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Romania District Olympiad 2001 - Grade IX

Source:

March 12, 2011
inequalitiesalgebra proposedalgebra

Problem Statement

Consider the equation x2+(a+b+c)x+λ(ab+bc+ca)=0x^2+(a+b+c)x+\lambda (ab+bc+ca)=0 with a,b,c>0a,b,c>0 and λ∈R\lambda\in \mathbb{R}. Prove that:
a)If λ≤34\lambda\le \frac{3}{4}, the equation has real roots. b)If a,b,ca,b,c are the side lengths of a triangle and λ≥1\lambda\ge 1, then the equation doesn't have real roots.
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