On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random point on the bottom disk.

Will the bottom disk be completely covered by the top disks with probability $1$?

We have a strong argument that if the bottom disk's area is less than $\pi$ then the probability that it will be covered is $1$, and if the bottom disk's area is greater than $\pi$ then the probability that it will be covered is less than $1$. But the boundary case seems to be more difficult.

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