Question 65
Let $A$ be a point in the Cartesian plane. Ann tells Bob a number $0 \lt a \le 1$ and he then moves $A$ rightward, leftward, upward or downward to a new position $A'$, $a$ distance apart from $A$. Next, Ann tells Bob a number $0 \lt a' \le 1$ and he moves $A'$ rightward, leftward, upward or downward to a new position $A''$, $a'$ distance apart from $A'$ and so on and so forth, as long as Ann wishes. At each step, Bob is free to choose which way to move the point, on one condition: Amongst every 100 consecutive moves, each of his four possible choices should have been made at least once. Ann's goal is to force Bob eventually choose a point strictly more than 100 distance away from $A$. Is it possible for Ann to achieve her goal?
Argentina, 2014
Argentina, 2014