Solution
= Solution
The fifty <disjoint sets> $\{1,100\},\{2,99\},\ldots,\{50,51\}$ form a <partition of a set> $\{1,\ldots,100\}$. 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. \b[Those two distinct elements sum to $101$.]