"am" or "will be"?

Yes. 'am' is correct, because at the time of the announcement you will be the winner.

Thank you.

But what I don't understand is why "Soon it will be announced that I will be the winner in our competition" is not correct, rhetorically awkward though.

"I will be the winner" would be suitable if the competition had not been held yet.

The statement "I am the winner" tells us that the competition has already been run and I assume this has been made clear in prior text.

context context context.

The contest began two weeks ago.

If you announce that Oristar will be the winner before the competition has been held, it might make people suspicious about whether it was fair.

If Oristar is actually the winner now, but the announcement has not yet been made, then it is correct to say that "it will be announced that I am the winnder".

Present forms for the future

http://www.englishgrammarsecrets.com/presentformsforfuture/menu.php

Soon it will be announced that I am the winner in our competition.

'am' indicates a state that exists. Whether that state of "winningness" actually does exist at the moment of speaking is not the sole determining factor. The speaker could be wanting to express a level of absolute certainty.

Because the victory is near, my girl.
Soon it will be announced that 'I am the winner' in our competition.

He is just projecting to his girl. Should have apostrophes.