[edparcell]

No need to brute-force. It's pretty easy to just eliminate possibilities:

The largest number must be larger than 7, else the largest possible sum of the three numbers is 21.

If the largest number is 8, then the other two must sum to 15, which is not possible since they must both be 7 or less.

So the largest number is 9, and the other two numbers sum to 14. The possibilities for those other two, with the smallest first are (5 and 9), (6 and 8), or (7 and 7). Clearly (5 and 9) would duplicate the original 9 (5+9+9), and (7 and 7) would duplicate the 7.

So the only possible solution is 6+8+9.