  A normal projective complex surface is called a rational homology projective
plane if it has the same Betti numbers with the complex projective plane
$\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane
with quotient singularities has at most 5 singular points. So far all known
examples have at most 4 singular points. In this paper, we prove that a
rational homology projective plane $S$ with quotient singularities such that
$K_S$ is nef has at most 4 singular points except one case. The exceptional
case comes from Enriques surfaces with a configuration of 9 smooth rational
curves whose Dynkin diagram is of type $ 3A_1 \oplus 2A_3$.
  We also obtain a similar result in the differentiable case and in the
symplectic case under certain assumptions which all hold in the algebraic case.
