USAMO 2017 — Days 1 and 2 Question 5

Find all real $c\gt 0$ for which the lattice points of $\mathbb Z^2$ can be labeled with positive integers so that only finitely many labels occur and any two distinct points with label $i$ are at distance at least $c^i$.

Proposed by Ricky Liu. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.