Question

Cho 3 số a, b, c  thỏa mãn: \(0< a\le b\le c\)

CMR: \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}\)

Answer 1

\(\Leftrightarrow ab^2+bc^2+ca^2\ge a^2b+b^2c+c^2a\)

\(\Leftrightarrow\left(c^2b-abc-b^2c+ab^2\right)+\left(ca^2+abc-ac^2-a^2b\right)\ge0\)

\(\Leftrightarrow b\left(c^2-ac-bc+ab\right)-a\left(c^2-ac-bc+ab\right)\ge0\)

\(\Leftrightarrow\left(b-a\right)\left(c^2-ac-bc+ab\right)\ge0\)

\(\Leftrightarrow\left(b-a\right)\left(c-b\right)\left(c-a\right)\ge0\) (luôn đúng do \(c\ge b\ge a>0\))