Denote by $B$ the set of all Bernoulli numbers.
Is it true that:
(a) For every rational number $r$ there exist a number $k=2^q$ (for some positive integer $q$) and numbers $a_1,\cdots,a_k\in B$ such that $$ r=a_1+\cdots+a_{\frac{k}{2}}-(a_{\frac{k}{2}+1}+\cdots+a_k). $$ Now, denote by $k(r)$ the least $k$ obtained from (a).
(b) The set of all $k(r)$, where $r$ runs over all rational numbers, is unbounded above.
Note. One can check that (a) is true when $r$ is an integer, by induction.