Greatest value.

[man111]

find The greatest value of $(a-x)(b-y)(c-z)(ax+by+cz)$. where $a,b,c$ are known positive quantities. and $(a-x),(b-y)$ and $(c-z)$ are also positive.

[venco]

А Вы не могли бы указать из какой олимпиады эти задачи?

[man111]

actually this is from text-book of Intermediate.

I have seen that this is a tough question. so I have post here.

[daogiauvang]

$$LRS= \frac{1}{abc} (a^2-ax)(b^2-bx)(c^2-cx)(ax+by+cz) \leq \frac{1}{abc} \frac{\left(a^2+b^2+c^2\right)^4}{256} $$

Знак "="

$$\iff x=\frac{3a^2-b^2-c^2}{4a}, y=\frac{3b^2-a^2-c^2}{4b}, z= \frac{3c^2-a^2-b^2}{4c}$$

[man111]

Thanks daogiauvang for nice solution.