Difference between revisions of "1993 OIM Problems/Problem 6"
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| − | Two non-negative integers <math>a</math> and <math>b</math> are ''mates'' if the decimal expression <math>a+b</math> consists only of zeros and ones. Let <math>A</math> and <math>B</math> be two infinite sets of non-negative integers, such that <math>B</math> is the set of all numbers that are ''mates'' of all the elements of <math>B</math>. | + | Two non-negative integers <math>a</math> and <math>b</math> are "''mates''" if the decimal expression <math>a+b</math> consists only of zeros and ones. Let <math>A</math> and <math>B</math> be two infinite sets of non-negative integers, such that <math>B</math> is the set of all numbers that are "''mates''" of all the elements of <math>B</math>. |
Prove that in one of the sets <math>A</math> or <math>B</math> there are infinitely many pairs of numbers <math>x</math>, and <math>y</math> such that <math>x-y = 1</math>. | Prove that in one of the sets <math>A</math> or <math>B</math> there are infinitely many pairs of numbers <math>x</math>, and <math>y</math> such that <math>x-y = 1</math>. | ||
Latest revision as of 14:22, 13 December 2023
Problem
Two non-negative integers 
and 
are "mates" if the decimal expression 
consists only of zeros and ones. Let 
and 
be two infinite sets of non-negative integers, such that is the set of all numbers that are "mates" of all the elements of .
Prove that in one of the sets or there are infinitely many pairs of numbers 
, and 
such that 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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