Proving an inequality using relationships between means

Question

Given three positive numbers a, b, c with the constraint:

$a+b+c=1$

prove the inequality

$\sqrt{{ab}\over{c+ab}}+\sqrt{{ac}\over{b+ac}}+\sqrt{{bc}\over{a+bc}}\le{3\over2}$

Answer 1

Using $a+b+c=1$, we can transform the L.H.S and use AM-GM. $$\sum_{cyc} \sqrt{\frac{ab}{c(a+b+c)+ab}} = \sum_{cyc} \sqrt{\frac{ab}{(c+a)(c+b)}} \le \sum_{cyc} \frac{1}{2} \left( \frac{a}{c+a} + \frac{b}{c+b} \right) = \frac{3}{2}$$