Balkan Mathematical Olympiad Shortlist

Denote the desired number by $t$. Consider the set $B$ consisting of all integers of the form $10a+b$, for $a=0,1,2,3,4,5$ and $b=1,2,3,4,5$, i.e. $$B=\{1,2,3,4,5,11,12,13,14,15,21,22,23,24,25,\dots ,51,52,53,54,55\}.$$ If $(10a_1+b_1)-(10a_2+b_2)=5$ then $10(a_1-a_2)=b_2-b_1+5$. Therefore, $b_2-b_1+5$ is divisible by $10$. On the other hand, $-4\le b_2-b_1\le 4$ applying $1\le b_2-b_1+5\le 9$, a contradiction. Since $B$ has cardinality $30$, we have $t\ge 30$.

Consider subset $B$ of $A$ having cardinality $t$. A pair $(m,n)$ is called good if $|m-n|=5$, $m\in B$ and $n\notin B$.

For every $m\in \{1,2,3,4,5,51,52,53,54,55\}$ there exists exactly one $n$ for $(m,n)$ to be a good pair. For all remaining values of $m$ there exist two values of $n$ for $(m,n)$ to be a good pair. Therefore, the number of good pairs is at least $10+2(t-10)=2t-10$. On the other hand, this number is at most $2(55-t)$ (since for every $n$ there exist at most two good pairs $(m,n)$). Thus, $$2t-10\le 110-2t$$ applying $t\le 30$. Therefore $t=30$.