Question 22
N5 (ROM), IMO3. For any positive integer $k$, $A_k$ is the subset of $\{k+1,k+2,\ldots,2k\}$ consisting of all elements whose digits in base $2$ contain exactly three $1$'s. Let $f(k)$ denote the number of elements in $A_k$.
(a) Prove that for any positive integer $m$, $f(k)=m$ has at least one solution.
(b) Determine all positive integers $m$ for which $f(k)=m$ has a unique solution.
(a) Prove that for any positive integer $m$, $f(k)=m$ has at least one solution.
(b) Determine all positive integers $m$ for which $f(k)=m$ has a unique solution.