IMO Shortlist 2017 problem C5

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Dodao/la: arhiva
Oct. 3, 2019

A hunter and an invisible rabbit play a game in the Eulidean plane. The hunter's starting point H_0 coincides with the rabbit's starting point R_0. In the n^{th} th round of the game (n \geq 1), the following happens.

(1) First the invisible rabbit moves secretly and unobserved from its current point R_{n-1} to some new point R_n with R_{n-1}R_n = 1.

(2) The hunter has a tracking device (e.g. dog) that returns an approximate position R'_n of the rabbit, so that R_nR'_n = 1.

(3) The hunter then visibly moves from point H_{n-1} to a new point H_n with H_{n-1}H_n = 1.

Is there a strategy for the hunter that guarantees that after 10^9 such rounds the distance between the hunter and the rabbit is below 100?