Grasshoppers on triangle lattice
Source: 2016 Ukraine TST
July 20, 2018
combinatoricscombinatorial geometry
Problem Statement
Let be an equilateral triangle of side . There are three grasshoppers sitting in , , . At any point of time for any two grasshoppers separated by a distance one of them can jump over other one so that distance between them becomes , are nonfixed positive integers. Let , be points on rays , such that , is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle . Find, in terms of , the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle during their jumps.)