Problem

Source: ELMO 2014 Shortlist N10, by Yang Liu

Tags: geometry, 3D geometry, modular arithmetic, number theory proposed, number theory



Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. Proposed by Yang Liu