I'm in a public speaking competition (school related) and I'm doing a solo acting event. I was wondering whether anyone knew any good monologues. (preferably from literature or a play). Anyone have any ideas?

You won't do better than Cookie's suggestion, it's inspiring. But if you don't fancy that, try John o'Gaunt's monologue from Act II of Richard II. You don't need to do the whole thing. This part is really inspiring.

This royal throne of kings, this sceptred isle,
This earth of majesty, this seat of Mars,
This other Eden, demi-paradise,
This fortress built by Nature for herself
Against infection and the hand of war,
This happy breed of men, this little world,
This precious stone set in the silver sea,
Which serves it in the office of a wall,
Or as a moat defensive to a house,
Against the envy of less happier lands,
This blessed plot, this earth, this realm this England.

There are a lot of good monologues in Macbeth as well this is my favorite:

MACBETH (Act 5, Scene 5, pg 2)                                               

She should have died hereafter.

There would have been a time for such a word.

Tomorrow, and tomorrow, and tomorrow,

Creeps in this petty pace from day to day

To the last syllable of recorded time,

And all our yesterdays have lighted fools

The way to dusty death. Out, out, brief candle!

Life’s but a walking shadow, a poor player

That struts and frets his hour upon the stage

And then is heard no more. It is a tale

Told by an idiot, full of sound and fury,

Signifying nothing