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Part of 2017 Iberoamerican

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Iberoamerican 2017 problem 1

Source: Iberoamerican 2017 p.1

9/20/2017
For every positive integer nn let S(n)S(n) be the sum of its digits. We say nn has a property PP if all terms in the infinite secuence n,S(n),S(S(n)),...n, S(n), S(S(n)),... are even numbers, and we say nn has a property II if all terms in this secuence are odd. Show that for, 1≤n≤20171 \le n \le 2017 there are more nn that have property II than those who have PP.
number theoryIberoamericaninternational competitions