2017 OIM Problems/Problem 1

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

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2017 OIM Problems/Problem 1

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