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Another quadratic!

Source: INMO 1998 Problem 5

October 7, 2005
quadraticsalgebrapolynomialinequalitiessum of rootsarea of a triangle

Problem Statement

Suppose a,b,ca,b,c are three rela numbers such that the quadratic equation x2−(a+b+c)x+(ab+bc+ca)=0 x^2 - (a +b +c )x + (ab +bc +ca) = 0 has roots of the form α+iβ\alpha + i \beta where α>0\alpha > 0 and β≠0\beta \not= 0 are real numbers. Show that (i) The numbers a,b,ca,b,c are all positive. (ii) The numbers a,b,c\sqrt{a}, \sqrt{b} , \sqrt{c} form the sides of a triangle.
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