KöMaL 2020 B5116

Let $a,b,c>0$ and $x,y,z\ge 0$. Prove that if $x+aby \le a(y+z)$, $y+bcz \le b(z+x)$, and $z+cax \le c(x+y)$, then $x=y=z=0$ or $a=b=c=1$.