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\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}}

Source: Moldova TST 2021

September 20, 2021
inequalities

Problem Statement

Positive real numbers aa, bb, cc satisfy a+b+c=1a+b+c=1. Show that a+1a+bc+b+1b+ca+c+1c+ab≥2a2+b2+c2.\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}. When does the equality take place?
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