The question doesn't say anything about these numbers other than that they're integers, so why not assume that 10 of them are:
-1, -2, -3, -4, -5, -6, -7, -8, -9, -10
Then you just need the 11th one to cancel those out. All the numbers above add up to -55, so make the last number +55. Now your sum is zero!
If you're going to have a sum equal to zero, you're going to need at least one positive number, but you don't necessarily need any more than that.
