Test of Mathematics Solution Subjective 80 - Inequality of squares

This is a Test of Mathematics Solution Subjective 80 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Problem

If $ {a, b, c}$ are positive numbers, then show that
$ \displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}} \ge {a + b + c}}$.

Solution

We know that $ {a, b, c > 0}$

L.H.S = $ {\displaystyle{{\frac{b^2 + c^2}{b + c}} + {\frac{c^2 + a^2}{c + a}} + {\frac{a^2 + b^2}{a + b}}}}$

$ {\ge}$ $ {\displaystyle{\frac{{\frac{b^2 + c^2 + 2bc}{b + c}} + {\frac{c^2 + a^2 + 2ac}{c + a}} + {\frac{a^2 + b^2 + 2ab}{a + b}}}{2}}}$ [ as $ {b + c}$, $ {c + a}$, $ {a + b > 0}$ & $ {x^2 + y^2 > 2xy}$ for all $ {x + y {\in} |R}$ ]

= $ {a + b + c}$
= R.H.S [ proved ]