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An operation in plane

Source: Greek national M.O. 2003, Final Round,problem 4

November 15, 2011
analytic geometrycombinatorics unsolvedcombinatorics

Problem Statement

On the set Σ\Sigma of points of the plane Π\Pi we define the operation ∗* which maps each pair (X,Y)(X, Y ) of points in Σ\Sigma to the point Z=X∗YZ = X * Y that is symmetric to XX with respect to Y.Y . Consider a square ABCDABCD in Π\Pi. Is it possible, using the points A,B,CA, B, C and applying the operation ∗* finitely many times, to construct the point D?D?
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