RO-TST 2005 Problem 13

a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$ , between $c_n$ and $c_{n+1}$ , the infinite sequence thus obtained does not represent the fractional part of a rational number.

b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$ .

Dan Schwartz