Question

[Lena]

Cho a,b,c,d > 0. Tìm min

S= a/b+c+d+b/a+c+d+c/a+b+d+d/a+b+b+c+d/a+a+c+d/b+a+b+d/c+a+b+c/d

Answer 1

[Kiệt Nguyễn]

Cô si lên:

\(S\ge8\sqrt[8]{\frac{abcd\left(b+c+d\right)\left(a+c+d\right)\left(a+b+d\right)\left(a+b+c\right)}{abcd\left(b+c+d\right)\left(a+c+d\right)\left(a+b+d\right)\left(a+b+c\right)}}=8\)

Answer 2

[zZz Cool Kid_new zZz]

๖²⁴ʱČøøℓ ɮøү 2к⁷༉ Liệu điểm rơi có xảy ra ???

Dùng \(\Sigma_{cyc}\) với \(\Pi_{cyc}\) cho nó lẹ nha,chớ mik nhác lắm:((

\(S=\Sigma_{cyc}\left(\frac{a}{b+c+d}+\frac{b+c+d}{a}\right)\)

\(=\Sigma_{cyc}\left(\frac{a}{b+c+d}+\frac{b+c+d}{9a}\right)+\Sigma_{cyc}\frac{8}{9}\cdot\frac{b+c+d}{a}\)

\(\ge8\sqrt[8]{\Pi_{cyc}\frac{a}{b+c+d}\cdot\Pi_{cyc}\frac{b+c+d}{9a}}+\frac{8}{9}\left(\frac{b}{a}+\frac{c}{a}+\frac{d}{a}+\frac{a}{b}+\frac{c}{b}+\frac{d}{b}+\frac{a}{c}+\frac{b}{c}+\frac{d}{c}+\frac{a}{d}+\frac{b}{d}+\frac{c}{d}\right)\)

\(\ge\frac{8}{3}+\frac{8}{9}\cdot12\left(use:\frac{x}{y}+\frac{y}{x}\ge2\right)\)

\(=\frac{40}{3}\)

Dấu "=" xảy ra tại a=b=c=d.

P/S:Viết tắt rồi mà vẫn dài:( Thử hỏi xem nếu ko viết thì sao ??