Solution
Guaranteed Pair Sum
Show the problem again
What is the minimum number of integers you must select from {1, 2, ..., 100} to guarantee that two of them sum to exactly 101?
Worked solution
The answer is 51. Partition {1, ..., 100} into the 50 pairs (1,100), (2,99), ..., (50,51), each summing to 101. Choosing 50 numbers can avoid every pair (take one from each), but by pigeonhole any 51 numbers must include both members of some pair. So 51 is necessary and sufficient.
Source: Classic pigeonhole-principle exercise in wide circulation. Statement written for AxiomIQ.