Unlimited sudden-death overtime has been used in the playoffs to determine winners since the early years of the NHL. The five minute overtime used in the regular season was introduced in 1983/84 season.
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Since the NCAA was not established until 1910, I would say that NCAA football did not begin until 1910. However, College Football was being played long before that, as early as 1869, though the game played then more closely resembled Rugby than modern American football. See the Wikipedia article on "college football", or just "American football", for more information.