Question

Cho x; y; z là các số dương nhỏ hơn 1 thỏa mãn x + y + z + 2\(\sqrt{xyz}\)= 1. Chứng minh rằng \(\sqrt{x\left(1-y\right)\left(1-z\right)}+\sqrt{y\left(1-x\right)\left(1-z\right)}+\sqrt{z\left(1-x\right)\left(1-y\right)}=1+\sqrt{xyz}\)

Answer 1

\(\sqrt{x\left(1-y\right)\left(1-z\right)}=\sqrt{x\left(yz-y-z+1\right)}=\sqrt{x\left(yz-y-z+x+y+z+2\sqrt{xyz}\right)}\)

\(=\sqrt{x\left(yz+x+2\sqrt{xyz}\right)}=\sqrt{x^2+2x\sqrt{xyz}+xyz}=\sqrt{\left(x+\sqrt{xyz}\right)^2}\)

\(=x+\sqrt{xyz}\)

Tương tự: \(\sqrt{y\left(1-x\right)\left(1-z\right)}=y+\sqrt{xyz}\) ; \(\sqrt{z\left(1-x\right)\left(1-y\right)}=z+\sqrt{xyz}\)

\(\Rightarrow VT=x+y+z+3\sqrt{xyz}=1-2\sqrt{xyz}+3\sqrt{xyz}=1+\sqrt{xyz}\) (đpcm)