Understand the problem

Let $a, b, c$ be the lengths of sides of a (possibly degenerate) triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

Source of the problem

Caucasus Mathematical Olympiad

Topic
Inequalities
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

The inequality is homogeneous, hence it might be a good idea to make the substitutions a=2R\sin A etc. The triangle inequality might also come in handy.
As the triangle is allowed to be degenerate, we have a+b\ge c. This implies ac+bc\ge c^2.
Summing, we get 2(ab+bc+ca)\ge a^2+b^2+c^2. It is possible to use the AM-GM inequality here to get the LHS of the inequality. Also, show that this implies 2(ab+bc+ca)\ge\frac{(a+b+c)^2}{2}.
Note that (a+b)\sqrt{ab}\ge 2ab by AM-GM. Hence, we get LHS\ge 2(ab+bc+ca)\ge a^2+b^2+c^2\ge\frac{(a+b+c)^2}{2}.

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