Question

[:vvv]

Cho x, y, z>0. Chứng minh rằng:

\(\dfrac{x}{x+2y+3z}+\dfrac{y}{y+2z+3x}+\dfrac{z}{z+2x+3y}\ge\dfrac{1}{2}\)

Answer 1

[Nguyễn Việt Lâm]

\(VT=\dfrac{x^2}{x^2+2xy+3zx}+\dfrac{y^2}{y^2+2yz+3xy}+\dfrac{z^2}{z^2+2zx+3yz}\)

\(VT\ge\dfrac{\left(x+y+z\right)^2}{x^2+y^2+z^2+5xy+5yz+5zx}=\dfrac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2+3\left(xy+yz+zx\right)}\ge\dfrac{\left(x+y+z\right)^2}{\left(x+y+z\right)^2+\left(x+y+z\right)^2}=\dfrac{1}{2}\)