5. The subset $X$ of the set $\{00,01, \cdots, 98,99\}$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain?
(52nd Moscow Olympiad problem)
5. For any $i, j \in\{0,1, \cdots, 9\}, X$ should contain either $ij$ or $ji$ (otherwise the sequence $ijyij \cdots$ would not meet the requirement). There are $10+C_{10}^{2}=55$ unordered pairs $(i, j)$, so $|X| \geqslant 55$.

On the other hand, if $X=\{ij \mid 0 \leqslant i \leqslant j \leqslant 9\}$, then $|X|=55$, and for any infinite sequence, let $i$ be the smallest digit it contains, and $j$ be the digit immediately following $i$, then $ij \in X$. Therefore, $X$ must contain at least 55 elements.