Maybe not the most elegant, but:

The lowest possible for x1 is 5, forcing (x2,x3)=(10,10).

Fix x1=6 then it's either (x2,x3)=(9,10) or (10,9)

Fix x2=7 then it's either (x2,x3)=(8,10) or (10,8) or (9,9)

As you can see there really aren't that many options if you continue in this manner.