Question

Cho các số x,y,z là các số dương thỏa mãn:

(x+y) (y+z) (x+z) =8xyz.CMR x = y = z

Answer 1

áp dụng bđt cosi cho 2 số dương, ta có:
\(x+y\ge2\sqrt{xy}\)

\(y+z\ge2\sqrt{yz}\)

\(x+z\ge2\sqrt{xz}\)

\(\Rightarrow\left(x+y\right)\left(y+z\right)\left(x+z\right)\ge2\sqrt{xy}.2\sqrt{yz}.2\sqrt{xz}=8\sqrt{x^2y^2z^2}=8xyz\)

Dấu "=" xảy ra khi \(x=y;y=z;x=z\Rightarrow x=y=z\)

\(\Rightarrow\left(x+y\right)\left(y+z\right)\left(x+z\right)=8xyz\Leftrightarrow x=y=z\left(đpcm\right)\)

Answer 2

ta có:

\(x+y\ge2\sqrt{xy}\)

\(y+z\ge2\sqrt{yz}\)

\(x+z\ge2\sqrt{xz}\)

\(\Rightarrow\left(x+y\right)\left(y+z\right)\left(x+z\right)\ge2\sqrt{yz}.2\sqrt{xz}\)

\(=8\sqrt{x^2y^2z^2}=8xyz\)

Ta thấy dấu = xảy ra khi:

\(x=y;y=z;x=z\Rightarrow x=y=z\)

\(\Rightarrow\left(x+y\right)\left(y+z\right)\left(x+z\right)=8xyz\Rightarrow x=y=z\left(đpcm\right)\)

Answer 3

Nguyễn Thanh HằngAkai HarumaPhạm Hoàng GiangAn TrầnAn Trịnh HữuPhạm Tú UyênBùi Thị VânGửi tin nhắn