The problem of determining the elastoplastic properties of a plate from an experimentally given relation between middle plane deflections and loads at some points on the plate is investigated. This problem leads to the inverse coefficient problem for the nonlinear differential equation of order four. The coefficient dependence of the nonlinear direct problem solution is studied. Then, reformulating the inverse problem as a minimization problem for a certain functional, the main existence theorem for the inverse problem considered is presented. The numerical algorithm and examples related to direct and inverse problems are considered.