USAMO 2017 Problem 5

Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $( x, y ) \in \mathbf{Z}^2$ with positive integers for which:

• only finitely many distinct labels occur, and
• for each label $i$ , the distance between any two points labeled $i$ is at least $c^i$ .

Proposed by Ricky Liu