I know we all get excited when we see writing contests that give us the opportunity to get our work in front of agents. I mean how cool would it be to get an agent through a contest? But I've got news for you. As great as contests are, they aren't the end all be all. They aren't the only path to finding an agent.
Contests are great learning experiences don't get me wrong. And you can meet loads of awesome writers, so by all means enter away. But make sure when you do, you don't agonize over them too much. Put all that energy into shining up your query, polishing your pages, absorbing all the writing tips you can, and building an awesome network of writers and critique partners.
Cause here's the thing, even if you make it into a contest and get in front of the agents, you might still come up empty. I've been there, all excited someone saw promise in my work only to get zero requests from the agents in the contest. It sucks, but rejection if part of the business.
So how is it possible that you busted your butt to get into a contest and then you get no requests? I have a theory, certainly subjectivity and marketability come into play here and what one person loves someone else might not, but sometimes it's a matter of who's involved. Sure you should be thoroughly researching the agents in contest before you enter to make sure a good chunk of the agents are looking for what you have. That said, when you're in a contest you're stuck with that limited set of agents.
On the other hand, when you query, you take control over who you submit to. You have the opportunity to show you've done your homework and to personalize your submission to each agent you send to. Many agents request pages in their submission guidelines and do read them which can also be to your benefit to submit more than just 250 words. (Yes, you should be able to hook someone in 250 words, but some agents are more forgiving than others if they see something there.) And you definitely get to submit your query, whereas in many contests you are limited to a short pitch. Sure, there's merit in being able to boil your book down to a sentence, but sometimes you need more real estate to really let your voice, and the unique aspects of your book shine.
The other advantage querying gives you is time. You can send them out in batches see how things go and reevaluate if needed. In contests it's a one shot deal. The submission windows are short, the contests don't usually last long, and you are often stuck showing agents what you sent in (although some contests have a rework/resubmit period it usually is on a tight deadline.) But when you query, you get rejections, and sometimes personalized feedback, and then you can take as much time as you need to perfect your submission package for the next round. Finding an agent is a marathon not a sprint so take every opportunity you can to perfect your work.
Another thing to consider about contests is there's a limited number of people selected. It's a firm number and a lot more people are going to enter than will actually get in. But if you query, an agent (or qualified intern) will see your work. And while they can only take on a limited number of clients, if they are open to queries, they are open to new clients. There's little limit to the number of requests they can make (other than reading time), and they can take on as many clients as they think they can adequately handle. So querying in many ways gives you a much larger window of opportunity.