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 
