The problem considered in this paper is given by the conditions: w=q+Mv+Nx, w≧0, v≧0, x≧0, v≦r, w T v=0 and (r−v) T x=0. Here q, r are n-vectors with r>0 and M, N are n by n matrices. We say that an n by 2n matrix (M, N) has the P-property if all of its “principal” minors are positive. The main reuslt is that (M, N) has the P-property if and only if the problem defined above has a unique solution for every q and every r>0. When M=N, this result reduces to the well-known existence and uniqueness theorem concerning the linear complementarity problem with a P-matrix. The above problem with (M, N) having the P-property includes as special cases, a strictly convex quadratic program with bounded variables and certain problems in structural mechanics.