We call a set on the real line superinvariant if for any stretching of the set by the transformation taking to there exists a translation such that the images of under and agree; i.e., for any there is a such that and for any there is a such that Determine all superinvariant sets.
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We call a set $S$ on the real line $\mathbb{R}$ superinvariant if for any stretching $A$ of the set by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0), a > 0$ there exists a translation $B,$ $B(x) = x+b,$ such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$ there is a $y \in S$ such that $A(x) = B(y)$ and for any $t \in S$ there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.
Izvor: Međunarodna matematička olimpijada, shortlist 1991
Školjka 
on the real line 
superinvariant if for any stretching 
of the set by the transformation taking 
to 
there exists a translation 
such that the images of 
agree; i.e., for any 
there is a 
such that 
and for any 
there is a 
such that 
Determine all superinvariant sets.