<s>[INST] Inequality. $\frac{ab+c}{a+b}+\frac{ac+b}{a+c}+\frac{bc+a}{b+c} \geq 2.$ Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that (using rearrangements inequalities, you can also view this exercise here, exercise number 3.1.8 )
$$\frac{ab+c}{a+b}+\frac{ac+b}{a+c}+\frac{bc+a}{b+c} \geq 2.$$
thanks. 
 [/INST] Using the identity $\displaystyle\frac{ab+c}{a+b}=\frac{ab+c^2}{a+b}+c,$ it suffices to check that
$$\sum_{cyc}\frac{ab}{a+b}+\sum_{cyc}\frac{c^2}{a+b}\geq a+b+c.$$
Note that the sequences $\{a^2,b^2,c^2\}$ and $\left\{\dfrac{1}{b+c},\dfrac{1}{c+a},\dfrac{1}{a+b}\right\}$ are similarly sorted, so that we obtain
$$\sum_{cyc}\frac{c^2}{a+b}\geq\sum_{cyc}\frac{a^2}{a+b},$$
Which, in accordance with $\dfrac{ab}{a+b}+\dfrac{a^2}{a+b}=a,$ leads us to our desired result. Equality occurs in the original inequality if and only if $a=b=c.$
$\Box$
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