This is my two cents to the competition idea. One of the most fun competitions i have participated was the Victor Ludus PS Vita competitions, dot2dots in facebook and riddles back when the Vita launched in 2012. The competition was organized trough twitter and included several competitions throughout the launch.
You could riddle or challenge (for example questions about FTC, the lore of cryptos in general, challenge to find some hidden message or detail in a youtube video, dot2dot challenge in Facebook, compare two pictures and find differences and many more) via the official FTC twitter to win 5,10 even 20FTC. These series of competitions could go trough the whole month of June actively and then a bit easier afterwards. In order to participate you would need to follow the official FTCtwitter and reply with the correct answer and hashtag #FTCRiddle/competition or whatever. First one/two/three to do this after the challenge is posted is/are the winner/s and could claim their prize by contacting via PM in twitter. The game could be advertised beforehand in btc-e trollbox, reddit, twitter, forum and other places. Between competitions you could invite people to check out the forum and the community and even join. This could be relatively cheap way to get people interested and new followers, get some publicity among the cryptocurrencies and let some of the newcomers chance to get some FTC.
Cryptorally has done this and gotten seemingly good publicity and followers and they have done this somewhat simple by just challenging people to promote their twitter in trollbox for some LTC. I think the secret is to try to do this in good taste so to speak and not to worry about giving out FTC “cheap”. The idea of this competition is simply raise awareness of FTC and the community and do this in a fun and even in a challenging way. You could even implement some of the other competitions from this thread to twitter after you have gotten some new followers (slogan for FTC, best use of the FTC logo…) To setup this competition you really don’t need anything you already haven’t got. Just your imagination is really the only limit in this. Hope this helps.
Capt. S
edit: after reading the original post again just wanted to clarify that you could obviously use USD or bigger amounts of FTC. This would be simply series of smaller contests. In this contest style it would be easier (or would it?) to use only FTC for prizes. Ideally get a lot of buzz and many different players and winners