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)