The exact "N Men Enter, One Man Leaves" can probably be traced back to Roman Gladiator fights, where at least one example from 264 BCE has three gladiators fight to the death. It's likely that prior to that, other cultures had events that were similar. Many gladiator fights were one on one though.
As far as what story or book first had the trope, it's hard to say. As far as I can tell, both The Running Man and The Long Walk by <PERSON> (writing as <PERSON>) were the first that had contestants attempting to outlast other people in a fight to the death (as it were) for the entertainment of the people (in The Running Man, the fight was one man against most of the world, and it was an extension of gameshows of the time. The Long Walk has more similarities with Battle Royal and The Hunger Games, but the contestants signed up for the event). Since both books were published in the 80s, it's likely that there are earlier examples.
Death Race 2000, a movie about a ... 72
erm, death race across the USA, again for entertainment, came out in 1975, and is based on the short story The Racer by <PERSON>. There's some similarities between the world presented in Death Race 2000 and worlds of Battle Royal and The Hunger Games. Finally, the episode "Fun and Games" from the original Outer Limits, aired in 1964, and had contestants from Earth fighting contestants from other planets to the death for the entertainment of an alien race (and the survival of all of Earth).
Doctor Who had a serial called The War Games, where humans from different time periods were dumped into an arena and fight to the death. The "winners" would then go on to be a super army for aliens, this is almost a "multiple people enter, one comes out" situation. On the other half of the trope, The Most Dangerous Game is about a hunter who hunts humans from 1924, and is perhaps the earliest (or at least, most famous) example of humans hunting humans as sport.
It's probably worth pointing out that the "reality show" version of this trope probably wouldn't appear much earlier than the 1950s, when TV shows and gameshows specifically became wildly popular, since the trope requires the audience to be familiar with the idea of people humiliating themselves for fame and money.
Found new information: The short story The Prize of Peril, by <PERSON> published in 1958, contains perhaps the earliest example of "deadly reality show", and was likely an inspiration for "The Running Man". A link to the complete story is on wikipedia. 322