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In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions \begin{equation*}\begin{cases} -(p(t)u'(t))'+q(t)u(t)=f(t,u(t)),\; 0<t<1,\\ \alpha_{1}u(0)-\beta_{1}u'(0)=\int_{0}^{1}u(\tau)\mathrm{d}\alpha(\tau),\\ \alpha_{2}u(1)+\beta_{2}u'(1)=\int_{0}^{1}u(\tau)\mathrm{d}\beta(\tau), \end{cases}\end{equation*} where $f(t,u)$ is allowed to be singular at $t=0,1$ and $u=0$. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.