Question

[Thị Thu Thúy Lê]

cho x,y,z la cac so duong thoa man \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=4\)

CMR:\(\frac{1}{2x+y+z}+\frac{1}{2y+x+z}+\frac{1}{2z+x+y}\le1\)

Answer 1

[Hoàng Lê Bảo Ngọc]

Áp dụng AM-GM ta có \(\frac{1^2}{x}+\frac{1^2}{x}+\frac{1^2}{y}+\frac{1^2}{z}\ge\frac{\left(1+1+1+1\right)^2}{2x+y+z}\)

hay \(\frac{2}{x}+\frac{1}{y}+\frac{1}{z}\ge\frac{16}{2x+y+z}\)

Tương tự : \(\frac{2}{y}+\frac{1}{x}+\frac{1}{z}\ge\frac{16}{2y+x+z}\) ; \(\frac{2}{z}+\frac{1}{x}+\frac{1}{y}\ge\frac{16}{2z+x+y}\)

Cộng theo vế : \(4\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\ge16\left(\frac{1}{2x+y+z}+\frac{1}{2y+x+z}+\frac{1}{2z+x+y}\right)\)

\(\Leftrightarrow\)\(16\left(\frac{1}{2x+y+z}+\frac{1}{2y+x+z}+\frac{1}{2z+x+y}\right)\le16\)

\(\Leftrightarrow\frac{1}{2x+y+z}+\frac{1}{2y+x+z}+\frac{1}{2z+x+y}\le1\)