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moving stones in lattice points, (a,b) ->(2a,b) or (a,2b), (a-b,b) or (a,b-a)

Source: 1995 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

January 27, 2020
lattice pointslatticecombinatorics

Problem Statement

Starting at (1,1)(1,1), a stone is moved in the coordinate plane according to the following rules: (i) From any point (a,b)(a,b), the stone can move to (2a,b)(2a,b) or (a,2b)(a,2b). (ii) From any point (a,b)(a,b), the stone can move to (a−b,b)(a-b,b) if a>ba > b, or to (a,b−a)(a,b-a) if a<ba < b. For which positive integers x,yx,y can the stone be moved to (x,y)(x,y)?
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