Difference between revisions of "2017 USAMO Problems/Problem 5"
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==Solution (INCOMPLETE)== | ==Solution (INCOMPLETE)== | ||
| − | For <math>c\le 1,</math> we can label every lattice point <math>1.</math> For <math>c\le 2^{1/4},</math> we can make a "checkerboard" labeling, i.e. label <math>(x, y)</math> with <math>1</math> if <math>x+y</math> is even and <math>2</math> if <math>x+y</math> is odd. One can easily verify that these labelings satisfy the required condition. Therefore, a labeling as desired exists for all <math>0 | + | For <math>c\le 1,</math> we can label every lattice point <math>1.</math> For <math>c\le 2^{1/4},</math> we can make a "checkerboard" labeling, i.e. label <math>(x, y)</math> with <math>1</math> if <math>x+y</math> is even and <math>2</math> if <math>x+y</math> is odd. One can easily verify that these labelings satisfy the required condition. Therefore, a labeling as desired exists for all <math>0 < c\le 2^{1/4}.</math> |
Revision as of 01:35, 3 May 2017
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 
