Prove that there are no positive real numbers $x$, $y$, $z$ such that $ (12x^2 + yz)(12y^2 + xz)(12z^2 + xy) = 2021x^2y^2z^2.$

Question

Prove that there are no positive real numbers $x, y, z$ such that : $(12x^2 + yz)(12y^2 + xz)(12z^2 + xy) = 2021x^2y^2z^2$. I managed to simplify it to $12(x^3z^3 + y^3z^3+x^3y^3) + xyz (x^3+y^3+z^3) = (73/3)(x^2y^2z^2$) but i still can't figure it out. Can someone help?

Answer 1

Following what you have left off, apply the AM-GM inequality to both factors : $x^3z^3+z^3y^3+x^3y^3 \ge 3x^2y^2z^2, x^3+y^3+z^3 \ge 3xyz$ and simplify the LHS you see that LHS > RHS . Done