CN-SMO 2022 Problem 8

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