IMO Shortlist 1991 problem 28


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April 2, 2012
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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.
Source: Međunarodna matematička olimpijada, shortlist 1991