MathDB

Tao plays the following game:given a constant

v>1

;for any positive integer

m

,the time between the

m^{th}

round and the

(m+1)^{th}

round of the game is

2^{-m}

seconds;Tao chooses a circular safe area whose radius is

2^{-m+1}

(with the border,and the choosing time won't be calculated) on the plane in the

m^{th}

round;the chosen circular safe area in each round will keep its center fixed,and its radius will decrease at the speed

v

in the rest of the time(if the radius decreases to

0

,erase the circular safe area);if it's possible to choose a circular safe area inside the union of the rest safe areas sometime before the

100^{th}

round(including the

100^{th}

round),then Tao wins the game.If Tao has a winning strategy,find the minimum value of

\biggl\lfloor\frac{1}{v-1}\biggr\rfloor

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