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Symmetric 3-var inequality

Source: 2007 Bulgarian Autumn Math Competition, Problem 12.3

March 17, 2022
Symmetric inequalitythree variable inequalityalgebrainequalities

Problem Statement

Find all real numbers rr, such that the inequality r(ab+bc+ca)+(3−r)(1a+1b+1c)≥9r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9 holds for any real a,b,c>0a,b,c>0.
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