A system of particles has its centre of mass at the origin. It...

The $x$-coordinate of the centre of mass of a system of particles is given by

$X_{cm} = \dfrac{\sum m_i x_i}{\sum m_i}$

Since the centre of mass is at the origin, $X_{cm} = 0$, which gives

$\sum m_i x_i = 0$

Since the mass $m_i$ of each particle is always positive, if there is a particle with a positive $x$-coordinate ($x_i > 0$), the term $m_i x_i$ will be positive. For the sum $\sum m_i x_i$ to be zero, there must be at least one term with a negative value. This implies that there must be at least one particle with a negative $x$-coordinate.

The number of particles or the total mass on either side of the origin need not be equal because the distance from the origin also determines the position of the centre of mass.