RU-ALLRUSSIAN 1971 Problem 14

Let $\operatorname {f}(\mathbf{x}, \mathbf{y}) = \mathbf{x}^{2} + \mathbf{x}\mathbf{y} + \mathbf{y}^{2}$ . Show that given any real $\mathbf{x}$ , $\mathbf{y}$ one can always find integers $\mathbf{m}, \mathbf{n}$ such that $\operatorname {f}(\mathbf{x} - \mathbf{m}, \mathbf{y} - \mathbf{n}) \leq = 1 / 3$ . What is the corresponding result if $\operatorname {f}(\mathbf{x}, \mathbf{y}) = \mathbf{x}^{2} + \mathbf{a}\mathbf{x}\mathbf{y} + \mathbf{y}^{2}$ with 0 $\leq \mathbf{a} \leq 2$ ?