Let A be any set of 19 distinct integers chosen from the arithmetic progression 1, 4, 7,..., 100.

Prove that there must be two distinct integers in A whose sum is 104.

You can safely pick the integers 1 and 52. You can also pick 1 integer from each of the 16 pairs that sums to 104: (4,100), (7,97), (10,94) ... (49,55). But then you are stuck. Your 19th pick must necessarily be the other half of a pair that sums to 104.