<s>[INST] For positive numbers a,b,c ,$abc=1$ prove$ \frac{1}{a+b} +\frac{1}{a+c}+\frac{1}{c+b} \leq\frac{3}{2}$ For positive numbers $a$,$b$,$c$ , with $abc=1$ prove $$\frac{1}{a+b} +\frac{1}{a+c}+\frac{1}{c+b}  \leq\frac{3}{2}$$
 [/INST] The problem as stated is false.  Take $a=b=0.1$, $c=100$.  $\frac{1}{a+b}=5$.
However, if $\min(a,b,c)\ge 1$ (for example, if they are integers), then it's true, by the other posted solution.
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