Yes you are. It's O(nlogn) where n is 5 - i.e. it's constant time (you can easily find the median by 4 comparisons to find the smallest of the 5, followed by 3 comparisons for the 2nd smallest, followed by 2 comparisons for the median - a total of 9 comparisons)
No it's not. Good design is about solving a problem, and you can't do that without understanding the problem. Just like you (and me! because I'm a compiler programmer, at heart) can't really solve a problem we don't understand. You can't come to me and tell me to build you a mobile app for your business, just because I know the ARM architecture... I need to understand your business first.
You know what's funny? We use the word "design" for software, and you wouldn't think that you could run a contest to pick the design of a software system. But nevertheless you think someone can design his communication strategy & brand identity by a contest, without spending the time to write down the requirements first.
To the OP: my advice is, just don't bother with a contest or a logo. Find someone (a friend) who is good at typography, and ask him to typeset the word "Tarsnap" for you in a way that looks good on that banner. Just use that as a logo, until you have the time & resources to hire someone to do a proper brand identity for you (because that's what you are asking for here, not just a logo)
This.
I can't believe nobody else pointed out that it's not n-squared.
(but, lots of people pointed out that complexity is really about reasoning not about memorisation - and you have to be good at reasoning in order to be a good programmer).
And this is not an "esoteric question" - you should really have a good grasp on complexity in order to understand the limitations of your system, and when you can use certain algorithms/data structures, and when you can't.
See this write-up: http://www.infoarena.ro/blog/numbers-everyone-should-know - but don't memorize the numbers, that would be pointless - the idea is to get a feel of why complexity is important.
(you'll notice for instance that between O(n!) and O(n^2) there is a 2-3 orders of magnitude difference in the acceptable input data size... so your confusion between factorial and n-squared is actually quite big)