$P=\frac{a^2+b^2+c^2}{2}+\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}=\frac{a^2+b^2+c^2}{2}+\frac{a^2+b^2+c^2}{abc}$$a^2+b^2+c^2\geq ab+bc+ac\Rightarrow P\geq \frac{a^2}{2}+\frac{1}{a}+\frac{b^2}{2}+\frac{1}{b}+\frac{c^2}{2}+\frac{1}{c}$
Có $\frac{a^2}{2}+\frac{1}{a}=\frac{a^2}{2}+\frac{1}{2b}+\frac{1}{2b}\geq \frac{3}{2}$
Từ đó suy ra GTNN