A frog trainer places one frog at each vertex of an equilateral triangle ABC 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 M and N be two points on the rays AB and AC, respectively, emanating from A, such that AM=AN=ℓ, where ℓ is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle AMN (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 AMN. (During the process, the frogs may leave the triangle AMN, only their nal positions are to be in that triangle.) combinatoricsEquilateral TriangleEquilateral