2017 IMO Problems/Problem 3
Problem
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, , and the hunter's starting point, , are the same. After rounds of the game, the rabbit is at point and the hunter is at point . In the nth round of the game, three things occur in order.
(i) The rabbit moves invisibly to a point such that the distance between and is exactly 1.
(ii) A tracking device reports a point to the hunter. The only guarantee provided by the tracking device is that the distance between and is at most 1.
(iii) The hunter moves visibly to a point such that the distance between and is exactly 1.
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 rounds she can ensure that the distance between her and the rabbit is at most 100?
Solution
Answer: No. There is no such strategy for the hunter. The rabbit will always “Win”
Proof: Suppose on the contrary that the answer is Yes. Therefore, there exists a strategy for the hunter to always “win” no matter how the rabbit moved or how the radar pinged. We will show that with bad luck from radar pings, the hunter cannot guarantee that the distance stays below after moves.
Let be the distance be the distance between the hunter and the rabbit after rounds. If the rabbit has won as all it needs to do is to move straight away from the hunter and the distance between the two will be kept at or above thereon. Now we tackle the other case, . We will show that whatever strategy the hunter follows, after rounds, the rabbit can increase by at least with lucky radar pings. This way, will reach in less than rounds, in which the rabbit wins. Suppose the hunter is at and the rabbit is at . Suppose the rabbit its location (this allow us to ignore all information from previous radar pings). Let be the line and let and be points which are unit away from and units away from , as in the figure below.
The rabbit’s plan is to simply choose one of the points or and hop rounds straight towards it. Since all hops stay within distance unit from , it is possible that all radar pings stay on . In particular, in this case, the hunter has no way of determining whether the rabbit is heading for or .
Looking at these pings, what can the hunter do? His best strategy is to go rounds straight to the right, ending up at point in the figure because the hunter will always be to the left of after the rounds, and if the hunter is above , then he will be further away from , and if he is below , then he will be further away from . In short, he can never be sure that the distance from him and the rabbit will be less than after these rounds. To estimate , we take as the midpoint of segment , we take as a point units to the right of , and define (Note that ). Then Where . In particular, , so Since and we assumed , this shows that . So ,as we claimed, with this list of radar pings, no matter what the hunter does, the rabbit might achieve The rabbit wins. ~Archieguan
See Also
2017 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |