Ant
Source: Iranian National Olympiad (3rd Round) 2002
October 13, 2006
geometry3D geometrycombinatorics proposedcombinatorics
Problem Statement
An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path.
a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic.
b) Prove that if if the ant starts from point and its path is periodic, then for each point if ant starts with this direction, then his path becomes periodic.