Library / /30 of 44

, 2025

Number theory Difficulty 5.0 AIME, harder Prove it United Kingdom

Consider the number $M = 99\ldots99$ which consists of several digit nines. A single division sign is placed between two adjacent digits of $M$ and the resulting calculation is evaluated to produce a whole number $N$ . In the case when $M$ has nine digits, how many possible values can $N$ take?

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.