Hint

Isn't that obvious? Take a set of 1008 numbers. Take the set of the other 1008 numbers. Use closure property of natural numbers BTW,my 2500th post yeah!!

Someone who can find a mistake in my solution? Cuz it does not seem to be convincing that it is an RMO problem

Math-wiz wrote: Isn't that obvious? Take a set of 1008 numbers. Take the set of the other 1008 numbers. Use closure property of natural numbers BTW,my 2500th post yeah!! Well, I thought the same thing on seeing the problem. Seems like some data is wrong or missing.

this is my source for the problem, the wording is as the source's

I agree with mathwiz, and I think this is pretty trivial. I mean just choose any arbitrary $1008$ numbers, their sum must be positive. The other $1008$ numbers must also sum to be a positive number, so adding the two sums yields a positive number.