The boundary value problem for harmonic functions in the exterior of certain cuts in the plane is considered. Two systems of cuts in the unit circumference are specified such that there are no common points in cuts. The Dirichlet boundary condition is specified on the inner side of each cut from the first system and on the outer side of each cut from the second system. The Neumann skew derivative boundary condition is specified on the other sides of all cuts from both systems. It is proved that the solution of this problem exists and is unique. The authors present the exact solution as an integral formula with undefined constants and a system of linear algebraic equations determining these constants uniquely. Some partial cases are considered. The solution is based on the reduction to the Riemann-Hilbert problem for complex analytic functions.