Can we prove that if, for a system of particles, the centre of mass is at origin then the summation of $xy$ over all the particles of the system is zero, where $x$ and $y$ are the respective coordinates of $i^\text{th}$ particle.
I tried to fix the any one coordinate, say $x$, but the summation of $y$ coordinates of the particles having that x coordinate is not zero, so I was unable to proceed further. Please help.