Question

Cho x,y,z > 2 và \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\). CMR: \(\left(x-2\right)\left(y-2\right)\left(z-2\right)\le1\)

Answer 1

Đặt \(\left(x-2,y-2.z-2\right)=\left(a,b,c\right)\) (a, b, c > 0).

Ta có: \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\Leftrightarrow\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}=1\)

\(\Leftrightarrow\frac{\left(ab+bc+ca\right)+4\left(a+b+c\right)+12}{abc+2\left(ab+bc+ca\right)+4\left(a+b+c\right)+8}=1\)

\(\Leftrightarrow abc+ab+bc+ca=4\).

Nếu \(abc>1\Rightarrow ab+bc+ca\ge3\sqrt[3]{\left(abc\right)^2}>3\Rightarrow abc+ab+bc+ca>4\) (vô lí).

Vậy \(\left(x-2\right)\left(y-2\right)\left(z-2\right)=abc\le1\).