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Proof this inequality on triangle

Source: 2019 Jozsef Wildt International Math Competition-W. 23

May 18, 2020

inequalitiesTriangle

Problem Statement

b

c

are the legs, and

a

is the hypotenuse of a right triangle, prove that

\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 5+3\sqrt{2}