Ta có: `1/(sqrt(yz)) + 2/(sqrt(xz)) + 3/(sqrt(xy)) = 9/(sqrt(xyz))`

`(sqrt(x) + 2sqrt(y) + 3sqrt(z))/(sqrt(xyz)) = 9/(sqrt(xyz))`

`sqrt(x) + 2sqrt(y) + 3sqrt(z) = 9`

Khi đó: `P = (sqrt(x) + 2sqrt(y) + 3)/(3sqrt(z) + 3) + (2sqrt(y) + 3sqrt(z) + 3)/(sqrt(x) + 3) + (3sqrt(z) + sqrt(x) + 3)/(2sqrt(y) + 3)`

`P + 3 = (sqrt(x) + 2sqrt(y) + 3)/(3sqrt(z) + 3) + 1 + (2sqrt(y) + 3sqrt(z) + 3)/(sqrt(x) + 3) + 1 + (3sqrt(z) + sqrt(x) + 3)/(2sqrt(y) + 3) + 1`

`P + 3 = (sqrt(x) + 2sqrt(y) + 3sqrt(z) + 6)/(3sqrt(z) + 3) + (sqrt(x) + 2sqrt(y) + 3sqrt(z) + 6)/(sqrt(x) + 3) + (sqrt(x) + 2sqrt(y) + 3sqrt(z) + 6)/(2sqrt(y) + 3)`

`P + 3 = 15/(3sqrt(z) + 3) + 15/(sqrt(x) + 3) + 15/(2sqrt(y) + 3)`

`P + 3 = 15 . (1/(3sqrt(z) + 3) + 1/(sqrt(x) + 3) + 1/(2sqrt(y) + 3))`

`>= 15 . ((1+1+1)^2)/(sqrt(x) + 2sqrt(y) + 3sqrt(z) + 3 + 3 + 3) = 15 . (3^2)/(9+9) = 15 . 9/18 = 15/2`

Khi đó: `P >= 15/2 - 3 = 15/2 - 6/2 = 9/2`

Dấu "=" xảy ra khi và chỉ khi `3sqrt(z) + 3 = sqrt(x) + 3 = 2sqrt(y) + 3`

`sqrt(x) = 2sqrt(y) = 3sqrt(z) = 9/3 = 3`

`{(x = 9),(y = 9/4),(z = 1):}`

Vậy, `min P = 9/2` khi và chỉ khi `(x;y;z) = (9;9/4;1)`