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frog jumping on lattice points

Source: Croatian MO 2004 2nd Grade P4

April 8, 2021
gamenumber theory

Problem Statement

A frog jumps on the coordinate lattice, starting from the point (1,1)(1,1), according to the following rules:
(i) From point (a,b)(a,b) the frog can jump to either (2a,b)(2a,b) or (a,2b)(a,2b); (ii) If a>ba>b, the frog can also jump from (a,b)(a,b) to (a−b,b)(a-b,b), while for a<ba<b it can jump from (a,b)(a,b) to (a,b−a)(a,b-a). Can the frog get to the point: (a) (24,40)(24,40); (b) (40,60)(40,60); (c) (24,60)(24,60); (d) (200,4)(200,4)?
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