If I knew that, say, I wanted a graphic of a piece of tar being snapped in half, I'd look for an artist and say "here's $500, please draw this for me". But given that I have no idea what I want, I figure the more people I invite to provide ideas the better.
That takes a lot of work--usually at least 30 to 40 hours for a good logo. Would you put in 40 hours of work for somebody on the off chance they might pay you?
You can rest assured of this as well: people will definitely submit to your contest--but none of them will be great designers, because all of them will be people who place the value of their work at somewhere around "maybe 500 dollars."
This would almost certainly be true if content participants had no connection to the business. In this case, participants my be considering their work to be pro-bono work with a value much greater $500.
It's possible that I'll find that someone has a nice idea but a poor implementation and award them the $500 but go off to hire an artist to draw the idea better.
You can just do what Steve Jobs did with Next: pick a designer you like, and leave it entirely up to him.
Ironically not because Jobs didn't have ideas on what he wanted but because the designer insisted:
"I asked him if he would come up with a few options, and he said, 'No, I will solve your problem for you and you will pay me. You don’t have to use the solution. If you want options go talk to other people.'"
Here is a piece of tar being snapped in half.. it just makes sense, you know you like it. You can have this one.. its yours.
If I were your designer-for-a-day, for 500 bucks i would do a few iterations once the concept is okayed to get it just right for you and give you all the svg / png / pdf or whatever other format you would like, and downscale to icon sizes etc.. but not until, in the immortal words of https://www.youtube.com/watch?v=cP_GOoBPKfQ (c/o mike monteiro;)
Would you run a code contest to have a software product built?