A set having the same mean, median, mode, and range

Question

Is it possible to have a set with the same mean, median, mode, and range?

If not, how can the following question be solved:

Set $H$ contains five positive integers such that the mean, median, mode, and range are all equal. The sum of the data is $25$.

Using the above information, indicate which one will be greater:

a) the smallest possible number in set $H$.

b) 6.

If I assume that all the elements in set $H$ are equal to $5$, it doesn't satisfy the conditions for range, as the range will become zero then.

Answer 1

The multiset $[3, 4, 5, 5, 8]$ will fit the bill.

You know, though, that even if you didn't have an example of a set on hand, the smallest element must be less than or equal to $5$ since the median is $5$ (since the mean is $5$).

Answer 2

Hint: If I allow non-integers and let the set contain duplicates (I think duplicates are allowed, though generally a set does not allow them. To have a mode you need duplicates), $\{2.5,5,5,5,7.5\}$ satisfies the constraints. Can you modify it to use only integers?