360.
Ta có:
S = \(\dfrac{a+b}{c}\)+\(\dfrac{b+c}{a}\)+\(\dfrac{c+a}{b}\)
S = \(\dfrac{a}{c}+\dfrac{b}{c}+\dfrac{b}{a}+\dfrac{c}{a}+\dfrac{c}{b}+\dfrac{a}{b}\)
S = \(\left(\dfrac{a}{c}+\dfrac{c}{a}\right)+\left(\dfrac{b}{c}+\dfrac{c}{b}\right)+\left(\dfrac{a}{b}+\dfrac{b}{a}\right)\)
Vì \(\dfrac{a}{c}+\dfrac{c}{a}\ge2\); \(\dfrac{b}{c}+\dfrac{c}{b}\ge2\); \(\dfrac{a}{b}+\dfrac{b}{a}\ge2\)
\(\Rightarrow\)S \(\ge\) 2 + 2 + 2
\(\Rightarrow\)S \(\ge\) 6
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