Question

Cho a,b,c dương. CMR

(a+b/ac).(b+c/ab).(c+a/bc)>=8

Answer 1

Đề là \(\left(a+\frac{b}{ac}\right)\left(b+\frac{c}{ab}\right)\left(c+\frac{a}{bc}\right)\ge8\) đúng ko bạn?

Ta có: \(a+\frac{b}{ac}\ge2\sqrt{\frac{ab}{ac}}=2\sqrt{\frac{b}{c}}\)

\(b+\frac{c}{ab}\ge2\sqrt{\frac{c}{a}}\) ; \(c+\frac{a}{bc}\ge2\sqrt{\frac{a}{b}}\)

Nhân vế với vế:

\(\left(a+\frac{b}{ac}\right)\left(b+\frac{c}{ab}\right)\left(c+\frac{a}{bc}\right)\ge8.\sqrt{\frac{abc}{abc}}=8\)

Dấu "=" xảy ra khi \(a=b=c=1\)