Shared alien birthdays

On the planet Maath, each year is exactly 20 days long. Find the smallest positive integer \(n\) such that if \(n\) randomly selected Maathians were in the same room, it would be more likely than not that two of them would share the same birthday. Assume that the birthday of each Maathian is independent and uniformly distributed among the 20 days of the year.


Inspired by this comic.
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