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find r if r (ab + bc + ca) + (3 r) (1/a+1/b+1/c) >=9 for a,b,c>0

Source: Puerto Rico TST 2010.6

September 14, 2021
inequalitiesalgebra

Problem Statement

Find all values ​​of r r 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) \ge 9 is true for a,b,ca,b,c arbitrary positive reals
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