\(a+bc=a\left(a+b+c\right)+bc=\left(a+b\right)\left(a+c\right)\)
\(\Rightarrow P=\dfrac{\left(a+b\right)\left(a+c\right)}{b+c}+\dfrac{\left(a+b\right)\left(b+c\right)}{a+c}+\dfrac{\left(a+c\right)\left(b+c\right)}{a+b}\)
Áp dụng BĐT Cô-si:
\(\dfrac{\left(a+b\right)\left(a+c\right)}{b+c}+\dfrac{\left(a+b\right)\left(b+c\right)}{a+c}\ge2\sqrt{\dfrac{\left(a+b\right)^2\left(a+c\right)\left(b+c\right)}{\left(b+c\right)\left(a+c\right)}}=2\left(a+b\right)\)
\(\dfrac{\left(a+b\right)\left(a+c\right)}{b+c}+\dfrac{\left(a+c\right)\left(b+c\right)}{a+b}\ge2\left(a+c\right)\)
\(\dfrac{\left(a+b\right)\left(b+c\right)}{a+c}+\dfrac{\left(a+c\right)\left(b+c\right)}{a+b}\ge2\left(a+b\right)\)
Cộng vế:
\(2P\ge4\left(a+b+c\right)=4\Rightarrow P\ge2\)
