Anyway, because my favorite subject is unsolicited writing advice, I'm going to impart my findings on you, o hapless reader. Here's my advice on beginnings: Start with a conflict, but not *the* conflict.
I think, especially for contests, that people are so hung up on trying to make that first page have EVERYTHING in it that you could possibly know, that they try to cram too much into it. Explosions! Fight scenes! Murder! Mayhem! The problem with this is that readers have no idea who these people are or why we should care about them. The other problem is that we need some sense of what a character's normal life is like before you start making it go crazy.
The way you do that is by introducing a character in a way that's relatatable and interesting, while avoiding excessive backstory and description. How do you do that? You have the character dealing with some fairly insignificant, but interesting, problem so that we can see how they handle it. This lets you use your "showing not telling" skills and gets us grounded into the story before it gets on. As long as we meet a character we like on Page 1, we'll keep reading. I promise.
Let's play the "famous book" game again! I have a very comfortable chihuahua in my lap so I can't get up to find the books for their actual first page, so I'm going by memory. Some of this is "first chapter" conflict in general, but you get the picture. Also some of these stories have prologues (or prologue-like first chapters), but since I already talked about those here I'll skip it and go to the beginning of the "real" first page.
Harry Potter & The Philosopher's Stone: We meet Harry, who lives in a cupboard under the stairs, which already gives us a pretty good sense of what kind of crappy family situation he's in. This is solidified by the pages that follow, where we see *exactly* what the Dursley's are like but more importantly how he reacts to them and how he's different from them.
Game of Thrones: We open with an execution, but what's important about the scene isn't the rolling head (of which we will see many more by the end of the book) but the way that Bran reacts to it -- and, even more important, the way that Ned Stark deals with it. You learn a lot about the family dynamic and the setting just by seeing it in action.
Neverwhere: After you skip the prologue, the story starts with a guy forgetting an important date -- and then trying to cover up for the lie. This whole setup is crucial to explaining Richard Mayhew's personality and the lifestyle he leads, including (especially) his fiancee and the trepidation there. Also it gives us an idea of just how important this dinner is and why his actions later are so bad.
Here's a few more tips:
- Don't open with a dream sequence; it sets up a false expectation and then says "Just kidding!" Don't troll your readers.
- Don't open with the MC just waking up. Not saying it can't be good, but everybody and their dog is doing it right now.
- If you open with dialogue, be sure to ground the reader *immediately* into who is talking (and why we care). Talking heads are a big turn-off.
- Don't open up with a description or the weather -- unless you can relate it to the character and do it in an engaging voice. In other words, a character struggling to get somewhere in the rain = works. A description of rain falling = doesn't work.
- Don't waste a really great conflict on a character we don't even know yet. Let us enjoy the MC just a little before you turn things upside down.