No such collection of integers can exist.
Let's say the integers are . Consider the sums for . If any of the sums is divisible by 10 we're done, so let's suppose none of them are. Then the possible remainders of the sums are 1, 2, ..., 9. Since there are 10 sums and 9 possible remainders, at least two of the sums, say and , with , have the same remainder, by the Pigeonhole Principle. Then is divisible by 10.