Any 5 distinct numbers chosen from the integers 1 to 8, result in at least two of them summing to 9.

We shall use the pigeonhole principle:

Consider the $4$ pairs of numbers $(1,8)$, $(2,7)$, $(3,6)$, and $(4,5)$. In each case, the sum of the two numbers is $9$. These pairs will serve as our "pigeon-holes". If we select $5$ distinct integers (i.e., the "pigeons") from the integers $1$ to $8$, inclusive -- then by the pigeonhole principle, at least two of them must be in the same pair. Since the $5$ integers chosen were distinct, we have found two that add up to 9.