They probably have to drop their output power too. There are a few stations, used to be called “clear channels”, that pretty much stay the same 24/7...they pretty much have the only station (at least in the US) at that particular frequency. WSB Atlanta, WLS Chicago, the one in N.O. and others, you can hear them almost from anywhere in the US at night if conditions are right.
From Wikipedia, here's a list of AM stations that are designated "clear channel", many of these are the originals and have been grandfathered. I don't believe the FCC issues 50,000 watt licenses these days anymore. (list includes Canadian and Mexican)