Question

Cho a,b,c > 1. Chứng minh

a² / b-1 + b² / c-1 + c² / a-1 >= 12

Answer 1

\(\dfrac{a^2}{b-1}+\dfrac{b^2}{c-1}+\dfrac{c^2}{a-1}\ge12\)

\(\Leftrightarrow\dfrac{a^2}{b-1}-4+\dfrac{b^2}{c-1}-4+\dfrac{c^2}{a-1}-4\ge0\)

\(\Leftrightarrow\dfrac{a^2-4b+4}{b-1}+\dfrac{b^2-4c+4}{c-1}+\dfrac{c^2-4a+4}{a-1}\ge0\)

\(a;b;c>1\Leftrightarrow a-1;b-1;c-1>0\)

\(\Leftrightarrow a^2-4b+4+b^2-4c+4+c^2-4a+4\ge0\)

\(\Leftrightarrow\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2\ge0\) (Đúng)

\("="\Leftrightarrow a=b=c=2\)