Difference between revisions of "2017 USAMO Problems/Problem 5"
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Let <math>\mathbf{Z}</math> denote the set of all integers. Find all real numbers <math>c > 0</math> such that there exists a labeling of the lattice points <math>( x, y ) \in \mathbf{Z}^2</math> with positive integers for which: only finitely many distinct labels occur, and for each label <math>i</math>, the distance between any two points labeled <math>i</math> is at least <math>c^i</math>. | Let <math>\mathbf{Z}</math> denote the set of all integers. Find all real numbers <math>c > 0</math> such that there exists a labeling of the lattice points <math>( x, y ) \in \mathbf{Z}^2</math> with positive integers for which: only finitely many distinct labels occur, and for each label <math>i</math>, the distance between any two points labeled <math>i</math> is at least <math>c^i</math>. | ||
| − | ==Solution | + | ==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\lt c\le 2^{1/4}.</math> | ||
Revision as of 01:34, 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 $0\lt c\le 2^{1/4}.$ (Error compiling LaTeX. Unknown error_msg)
