MathDB

Mount inequality erupts in El Salvador

Source: 2023 Centroamerican and Caribbean Math Olympiad, P3

July 25, 2023

inequalities

Problem Statement

Let

a,\ b

and

c

be positive real numbers such that

a b+b c+c a=1

. Show that

\frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} .