RU-ALLRUSSIAN 2015 Problem 115

An immortal flea jumps on whole points of the number line, beginning with $0$ . The length of the first jump is $3$ , the second $5$ , the third $9$ , and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$ . The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?