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sum (ab+ 1)/(a^2 + ca + 1) >= 3/2 for triangle sidelengths

Source: Switzerland - 2016 Swiss MO Final Round p2

December 30, 2022

algebrainequalitiesGeometric Inequalities

Problem Statement

Let

a, b

and

c

be the sides of a triangle, that is:

a + b > c

b + c > a

and

c + a > b

. Show that:

\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32