Problem 3

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

A_{n-1}

and the hunter is at

B_{n-1}

. In the

n

th round, three things occur in order: (i) the rabbit moves invisibly to a point

A_n

such that the distance between

A_{n-1}

and

A_n

is exactly

1

; (ii) a tracking device reports a point

P_n

to the hunter, with the only guarantee that the distance between

P_n

and

A_n

is at most

1

; (iii) the hunter moves visibly to a point

B_n

such that the distance between

B_{n-1}

and

B_n

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

10^9

rounds she can ensure that the distance between her and the rabbit is at most

100

Step 7 of 9: Each block increases squared distance

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y^2=d^2+\varepsilon(400-2d)>d^2+\frac12

Detailed analysis

Substitute $\varepsilon^2+1=400\varepsilon$ into $y^2=1+(d-\varepsilon)^2$ to get $y^2=d^2+\varepsilon(400-2d)$ . Because $d\le100$ , we have $400-2d\ge200$ , and because $\varepsilon>1/400$ , it follows that $y^2>d^2+1/2$ . Thus, regardless of the hunter's moves, the rabbit can choose one of the indistinguishable routes so that after $200$ rounds the new squared distance is greater than the old one by $1/2$ .