Given the numbers 1,2,3,4,…,n can you always erase one number so that the mean of the remaining numbers is an integer?
If n is odd, erase the middle number.
If n is even, erase 1 or n.
In general, with any n numbers (not necessarily consecutive integers), calculate the sum mod (n-1), and you can succeed if one or more of those numbers have that value mod (n-1). Clearly, with n consecutive integers, at least one will equal mod (n-1) the total mod (n-1).