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Inequality between the lengths of the sides of a triangle

Source: Poland Math Olympiad 1993 Round 1 #3

June 3, 2023

inequalities

Problem Statement

Prove that if

a,b,c

are the lengths of the sides of a triangle, then

\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leq \frac{1}{a+b-c}+\frac{1}{c+a-b}+\frac{1}{b+c-a}