## IMO '94 P3

Discussion on International Mathematical Olympiad (IMO)
Abdullah Al Tanzim
Posts: 22
Joined: Tue Apr 11, 2017 12:03 am

### IMO '94 P3

For any positive integer $k$, let $f(k)$ be the number of elements in the set ${k+1,k+2,...,2k}$ whose base $2$ representation has precisely three $1s$.

(a) Prove that for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k)=m$.
(b) Determine all positive integer $m$ for which there exists exactly one $k$ with $f(k)=m$.
Everybody is a genius.... But if you judge a fish by its ability to climb a tree, it will spend its whole life believing that it is stupid - Albert Einstein

Abdullah Al Tanzim
Posts: 22
Joined: Tue Apr 11, 2017 12:03 am