one frog at each vertex of an equilateral triangle, frogs jumping
Source: RMM Shortlist 2016 C2
July 4, 2019
combinatoricsEquilateral TriangleEquilateral
Problem Statement
A frog trainer places one frog at each vertex of an equilateral triangle of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let and be two points on the rays and , respectively, emanating from , such that , where is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle (inside or on the boundary), and no more jumps are performed.
Determine the number of final positions the three frogs may reach in the triangle . (During the process, the frogs may leave the triangle , only their nal positions are to be in that triangle.)