Đặt \(a=y+z,b=z+x,c=x+y\) (với a > 0, b > 0, c> 0)
\(\Rightarrow x=\frac{b+c-a}{2},y=\frac{c+a-b}{2},z=\frac{a+b-c}{2}\). Ta có
\(VT\) \(P=\frac{25\left(b+c-a\right)}{2a}+\frac{4\left(c+a-b\right)}{2b}+\frac{9\left(a+b-c\right)}{2c}\)\(=\left(\frac{25b}{2a}+\frac{4a}{2b}\right)+\left(\frac{25c}{2a}+\frac{9a}{2c}\right)+\left(\frac{4c}{2b}+\frac{9b}{2c}\right)-19\ge10+15+6-19=12\)
Đẳng thức xảy ra khi và chỉ khi
\(\left\{{}\begin{matrix}5b=2a\\5c=3a\end{matrix}\right.\)
\(\Rightarrow5b+5c=5a\Rightarrow x=0\left(vôlis\right)\)
Vậy BĐT P đúng