The "mode" exists.

Meaning there is a number which appears most often.

Thus, there are three possibilities:

1)Mode is 25 ( .

2)Mode is 19 ( .

3)Mode is 39 ( ).

It cannot be case #3 because then the mean is 29.2 a violation of initial conditions of problem.

If #2 then the mean must be 19.

Meaning,

Checking the mode we find that this works.

If #1 then the mean must be 25.

Meaning,

But the problem says it is not amongs the numbers.

Thus, we have determined the unique numbers the work.