MathNet 1971 — Part 01 Question 15

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