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4 frogs sit on vertices of a square, jumping

Source: TOT 397 1993 Autumn A S5 - Tournament of Towns

June 12, 2024
combinatoricsgeometrycombinatorial geometry

Problem Statement

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)?
(A Andzans)