Call admissible a set of integers that has the following property: If (possibly ) then for every integer . Determine all pairs of nonzero integers such that the only admissible set containing both and is the set of all integers.
Proposed by Warut Suksompong, Thailand
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Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
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.
Proposed by Warut Suksompong, Thailand
Source: Međunarodna matematička olimpijada, shortlist 2012
Školjka 
of integers that has the following property:
(possibly 
) then 
for every integer 
.
of nonzero integers such that the only admissible set containing both 
and 
is the set of all integers.