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In this paper we analyse for the first time the bending vibration of a nanoplate with an attached mass using the strain gradient elasticity theory for homogeneous Lame material, under Kirchhoff-Love's kinematical assumptions. The exact eigenvalues of the nanoplate vibrating with an attached mass are obtained for a general case, and an approximate closed form expression is provided if the intensity of the mass is small with respect to the total mass of the nanoplate. The inverse problem of identifying a point mass attached on a simply supported rectangular nanoplate from a selected minimal set of resonant frequency data is also considered. We show that if the point mass is small, then the position of the point mass and mass size can be determined by means of closed form expressions in terms of the changes induced by the point mass on the first three resonant frequencies. The identification procedure has been tested on an extended series of numerical simulations, varying the scale parameter of the nanoplate's material and the position and size of the point mass.