Let ABC be an equilateral triangle of side 1. There are three grasshoppers sitting in A, B, C. At any point of time for any two grasshoppers separated by a distance d one of them can jump over other one so that distance between them becomes 2kd, k,d are nonfixed positive integers. Let M, N be points on rays AB, AC such that AM=AN=l, l is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle AMN. Find, in terms of l, the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle AMN during their jumps.) combinatoricscombinatorial geometry