Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 3 / c

The fifty disjoint sets $\{1,100\},\{2,99\},\dots ,\{50,51\}$ form a partition of a set $\{1,\dots ,100\}$. If a chosen subset met each pair at most once, it would contain at most fifty elements. The pigeonhole principle therefore forces a $51$-element subset to contain both elements of at least one pair. Those two distinct elements sum to $101$.