Difference between revisions of "2017 OIM Problems/Problem 1"

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== Problem ==
 
== Problem ==
For each positive integer <math>n</math>, let <math>S(n)</math> be the sum of its digits. We say that <math>n</math> has the property <math>E</math> if the terms of the infinite sequence <math>n, S(n), S(S(n)), S(S(S(n))), \cdots </math>, are all even, and we say that <math>n</math> has property <math>O</math> if the terms of this sequence are all odd. Show that among all the positive integers <math>n</math> such that <math>1 \le n \le 2017</math> there are more who have property "O" than those who have property <math>E</math>.
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For each positive integer <math>n</math>, let <math>S(n)</math> be the sum of its digits. We say that <math>n</math> has the property <math>E</math> if the terms of the infinite sequence <math>n, S(n), S(S(n)), S(S(S(n))), \cdots </math>, are all even, and we say that <math>n</math> has property <math>O</math> if the terms of this sequence are all odd. Show that among all the positive integers <math>n</math> such that <math>1 \le n \le 2017</math> there are more who have property <math>O</math> than those who have property <math>E</math>.
  
 
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
 
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Latest revision as of 13:36, 14 December 2023

Problem

For each positive integer $n$, let $S(n)$ be the sum of its digits. We say that $n$ has the property $E$ if the terms of the infinite sequence $n, S(n), S(S(n)), S(S(S(n))), \cdots$, are all even, and we say that $n$ has property $O$ if the terms of this sequence are all odd. Show that among all the positive integers $n$ such that $1 \le n \le 2017$ there are more who have property $O$ than those who have property $E$.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions