Question

Cho các số dương a,b,c CMR

\(\frac{a^4}{b^3(c+2a)}+\frac{b^4}{c^3(a+2b)}+\frac{c^4}{a^3(b+2c)}\ge 1\)

@Akai Haruma

Answer 1

Dựa theo cách của Akai Haruma,bài này # bài của bạn ý nên t làm luôn: \(\dfrac{a^4}{b^3\left(c+2a\right)}+\dfrac{b^4}{c^3\left(a+2b\right)}+\dfrac{c^4}{a^3\left(b+2c\right)}\)

\(=\dfrac{\dfrac{a^4}{b^2}}{bc+2ab}+\dfrac{\dfrac{b^4}{c^2}}{ac+2bc}+\dfrac{\dfrac{c^4}{a^2}}{ab+2ac}\)

\(\ge\dfrac{\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\right)^2}{3\left(ab+bc+ac\right)}\ge\dfrac{\left[\dfrac{\left(a+b+c\right)^2}{a+b+c}\right]^2}{3\left(ab+bc+ac\right)}\ge\dfrac{\left(a+b+c\right)^2}{3\left(ab+bc+ac\right)}\ge\dfrac{3\left(ab+bc+ac\right)}{3\left(ab+bc+ac\right)}=1\)