Lời giải:
Ta có `:` `xy+z=xy+z(x+y+z)=(x+z)(y+z)`
Do đó `:`
`\sqrt[(xy)/(xy+z)]=\sqrt[{xy}/{(x+z)(y+z)}]<=1/2(x/(x+z)+y/(y+z))``text( (1))`
Tương tự ta cũng có `:`
`\sqrt[(zy)/(zy+x)]=\sqrt[{zy}/{(x+y)(x+z)}]<=1/2(y/(x+y)+z/(x+z))``text( (2))`
`\sqrt[(zx)/(zx+y)]=\sqrt[{zx}/{(z+y)(x+y)}]<=1/2(z/(z+y)+x/(x+y))``text( (3))`
Cộng `(1),(2),(3)` theo từng vế ta có `:`
`\sqrt[(xy)/(xy+z)]+\sqrt[(zy)/(zy+x)]+\sqrt[(zx)/(zx+y)]<=3/2`
Vậy `:` `\sqrt[(xy)/(xy+z)]+\sqrt[(zy)/(zy+x)]+\sqrt[(zx)/(zx+y)]<=3/2` luôn đúng.