Question

Cho a, b, c > 0 và a + b + c + ab + bc + ca = 6abc. Chứng minh rằng

\(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\) ≥ 3

Answer 1

\(a+b+c+ab+bc+ca=6abc\Leftrightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=6\)

Đặt \(\left(\frac{1}{a};\frac{1}{b};\frac{1}{c}\right)=\left(x;y;z\right)\Rightarrow x+y+z+xy+yz+zx=6\)

Ta cần chứng minh: \(x^2+y^2+z^2\ge3\)

Thật vậy:

\(x^2+1+y^2+1+z^2+1\ge2x+2y+2z\)

\(2\left(x^2+y^2+z^2\right)\ge2\left(xy+yz+zx\right)\)

Cộng vế với vế:

\(3\left(x^2+y^2+z^2\right)+3\ge2\left(x+y+z+xy+yz+zx\right)\)

\(\Leftrightarrow3\left(x^2+y^2+z^2\right)+3\ge12\)

\(\Rightarrow x^2+y^2+z^2\ge3\)

Dấu "=" xảy ra khi \(\left(x;y;z\right)=\left(1;1;1\right)\) hay \(\left(a;b;c\right)=\left(1;1;1\right)\)