2017 USAMO Problems/Problem 5

Problem

Let denote the set of all integers. Find all real numbers such that there exists a labeling of the lattice points with positive integers for which: only finitely many distinct labels occur, and for each label , the distance between any two points labeled is at least .

Solution (INCOMPLETE)

For we can label every lattice point For we can make a "checkerboard" labeling, i.e. label with if is even and if is odd. One can easily verify that these labelings satisfy the required condition. Therefore, a labeling as desired exists for all $0\lt c\le 2^{1/4}.$ (Error compiling LaTeX. Unknown error_msg)