KöMaL 2024 A889

Let $W, A, B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent.

• For all $x$, $y$, $z\geq0$ satisfying $x+y\leq z+W$, $x+z\leq y+W$, $y+z\leq x+W$ we have $Axyz+B\geq x^{2}+y^{2}+z^{2}$.
• $B\geq W^{2}$ and $AW^{3}+B\geq3W^{2}$.