Question 6
Find all real $x,y,z\ge1$ such that
$$\min\bigl(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz}\bigr)=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$
Proposed by Titu Andreescu (as credited in the supplied compilation).
$$\min\bigl(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz}\bigr)=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$
Proposed by Titu Andreescu (as credited in the supplied compilation).