Question

[Châu Anh Minh]

cho 3 số dương a,b,c thỏa mãn ab+bc+ca=3.Chứng minh rằng :(a+b)(b+c)(c+a)>=8

Answer 1

[Nguyễn Việt Lâm]

Ta có:

\(\left(a+b\right)\left(b+c\right)\left(c+a\right)=\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)

\(=\left(a+b+c\right)\left(ab+bc+ca\right)-\sqrt[3]{abc}.\sqrt[3]{ab.bc.ca}\)

\(\ge\left(a+b+c\right)\left(ab+bc+ca\right)-\dfrac{1}{3}\left(a+b+c\right).\dfrac{1}{3}\left(ab+bc+ca\right)\)

\(=\dfrac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)

Do đó:

\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\dfrac{8}{9}.3.\left(a+b+c\right)\ge\dfrac{8}{3}\sqrt{3\left(ab+bc+ca\right)}=8\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=1\)