To quote Tommy Jarrell, "Anybody who enters a banjo contest deserves to lose."
There are as many forms of competition as there are banjo players. At Galax, VA (2nd week of August) you play ONE tune, as many times as you can play in 2 min 30 seconds, and get off the stage, (over 100 entries) and they pin the top ten places at the end of the week. At Clifftop (the weekend before) you play 1 tune; they pick the top 5 finalists, those play 2 tunes each. Generally banjo contests will have one contest for Bluegrass Banjo and one for Old-time Banjo, which usually means clawhammer. In Morris CT and Roxbury CT play one tune, they announce the winner at the end. The Florida Old-time Music Championships -- you play TWO tunes in the first round, they pick five finalists, and those finalists play THREE tunes in second round. It really kinda depends on how long the contests as a whole are to last-- you want to fill up an evening for the spectators and give them their money's worth (not quite a joke)

One of the things I love about the smaller festivals is the opportunity to meet and make music w the other players. One year at FL they held the "string band contest" between the 1st round banjo and finals. . . the five finalists were all practicing and jamming and VGO said "hey, why don't we go up as a band?" so we went up and played two tunes, almost in Dixieland style (each person playing melody, in turn, with the others playing soft backup) the judges disqualified us for not having a fiddle. This should give you an idea of how seriously those contests are taken. . . Clifftop and Galax, OTOH, have serious prize money or serious bragging rights.