RU-ALLRUSSIAN 1981 Problem 16

The positive reals $\mathbf{x}$ , y satisfy $\mathbf{x}^{3} + \mathbf{y}^{3} = \mathbf{x} - \mathbf{y}$ . Show that $\mathbf{x}^{2} + \mathbf{y}^{2} < 1$ .