2012 IMO Shortlist N1

Call admissible a set $A$ of integers that has the following property: $$\text { If } x, y \in A \text { (possibly } x=y \text { ) then } x^{2}+k x y+y^{2} \in A \text { for every integer } k \text {. }$$ Determine all pairs $m, n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.

称可接受的整数集 $A$ 具有以下属性： \text { 如果 } x, y \in A \text { (可能 } x=y \text { )则 } x^{2}+k x y+y^{2} \in A \text { 对于每个整数 } k \text {。 确定所有非零整数对 $m, n$，使得包含 $m$ 和 $n$ 的唯一可接受的集合是所有整数的集合。