Pairs of numbers

A collection can have $n\le 12$ : just pick the first $n$ .

Note $3$ and $8$ are the only listed numbers who aren't part of a pair of numbers listed that sums to $121$ . the rest of the numbers ( $13,18,\ldots,103,108$ ) form $10$ disjoint pairs that sum to $121$ . By the pigeonhole principle, if we pick more than $12$ numbers from the list, we must pick two numbers who sum to $121$ . Therefore, the answer is $\boxed{13}$ .