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Omni-MATH / A hunter and an invisible rabbit play a game in the Euclidean plane. The…

Problem

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problem

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, A0,A_0, and the hunter's starting point, B0B_0 are the same. After n−1n-1 rounds of the game, the rabbit is at point An−1A_{n-1} and the hunter is at point Bn−1.B_{n-1}. In the nthn^{\text{th}} round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point AnA_n such that the distance between An−1A_{n-1} and AnA_n is exactly 1.1. [*]A tracking device reports a point PnP_n to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between PnP_n and AnA_n is at most 1.1. [*]The hunter moves visibly to a point BnB_n such that the distance between Bn−1B_{n-1} and BnB_n is exactly 1.1. [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after 10910^9 rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]
Plain-text mathematical notation (without MathML)
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, A₀, and the hunter's starting point, B₀ are the same. After n−1 rounds of the game, the rabbit is at point A_(n−1) and the hunter is at point B_(n−1). In the n^(th) round of the game, three things occur in order:

[list=i]
[*]The rabbit moves invisibly to a point A_(n) such that the distance between A_(n−1) and A_(n) is exactly 1.

[*]A tracking device reports a point P_(n) to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between P_(n) and A_(n) is at most 1.

[*]The hunter moves visibly to a point B_(n) such that the distance between B_(n−1) and B_(n) is exactly 1.
[/list]
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after 10⁹ rounds, she can ensure that the distance between her and the rabbit is at most $100?$

[i]
Original LaTeX notation
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order:

[list=i]
[*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$

[*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$

[*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$
[/list]
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$

[i]

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Source and history

task_id
cc77cbe1-c706-55a5-bc97-f5cf3a6f2cbf
task_key
test--cc77cbe1-c706-55a5-bc97-f5cf3a6f2cbf
task_revision_id
1
config / split
/ test

Official source

initial import