Question 3
A hunter and an invisible rabbit play a game in the plane. The rabbit and hunter start at points $A_0 = B_0$. In the $n$th round of the game ($n \ge 1$), three things occur in order:
(i) The rabbit moves invisibly from $A_{n-1}$ to a point $A_n$ such that $A_{n-1} A_n = 1$.
(ii) The hunter has a tracking device (e.g. dog) which reports an approximate location $P_n$ of the rabbit, such that $P_n A_n \le 1$.
(iii) The hunter moves visibly from $B_{n-1}$ to a point $B_n$ such that $B_{n-1} B_n = 1$.
Let $N = 10^9$. Can the hunter guarantee that $A_NB_N\le100$?
(i) The rabbit moves invisibly from $A_{n-1}$ to a point $A_n$ such that $A_{n-1} A_n = 1$.
(ii) The hunter has a tracking device (e.g. dog) which reports an approximate location $P_n$ of the rabbit, such that $P_n A_n \le 1$.
(iii) The hunter moves visibly from $B_{n-1}$ to a point $B_n$ such that $B_{n-1} B_n = 1$.
Let $N = 10^9$. Can the hunter guarantee that $A_NB_N\le100$?