Đặt \(\sqrt{x}=a;\sqrt{y}=b;\sqrt{z}=c\left(a;b;c>0\right)\) và \(p=a+b+c;q=ab+bc+ca;r=abc\)
Thì \(a^2+b^2+c^2+2=a^2b^2c^2\Leftrightarrow p^2-4q+2=r^2-2q\)
Cần chứng minh: \(a^2+b^2+c^2+6\ge2\left(ab+bc+ca\right)\Leftrightarrow p^2-2q+6\ge2q\)
Nếu \(q\le6\): Có \(p^2\ge3q\) nên ta chứng minh \(q+6\ge2q\Leftrightarrow q\le6\) (đúng)
Nếu \(q>6\) mình chưa nghĩ ra.
@Akai Haruma cô có cách nào khác hoặc cách nào cho trường hợp q > 6 không cô?
\(x+y+z+2=xyz\)
\(\Leftrightarrow2x+2y+2z+xy+yz+zx+3=\left(x+1\right)\left(y+1\right)\left(z+1\right)\)
\(\Leftrightarrow\left(x+1\right)\left(y+1\right)+\left(y+1\right)\left(z+1\right)+\left(z+1\right)\left(x+1\right)=\left(x+1\right)\left(y+1\right)\left(z+1\right)\)
\(\Leftrightarrow\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=1\)
\(\Leftrightarrow\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}=2\)
\(\Rightarrow2=\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}\ge\frac{\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)^2}{x+y+z+3}\)
\(\Leftrightarrow2x+2y+2z+6\ge\left(\sqrt{x}+\sqrt{y}+\sqrt{z}\right)^2\)
\(\Leftrightarrow2x+2y+2z+6\ge x+y+z+2\sqrt{xy}+2\sqrt{yz}+2\sqrt{zx}\)
\(\Leftrightarrow x+y+z+6\ge2\left(\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\right)\)
Dấu "=" xảy ra khi \(x=y=z=2\)