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Progetto Olimpiadi di Matematica GARA di SECONDO LIVELLO TRIENNIO, 2003

Algebra Difficulty 4.8 AIME Prove it Italy

Problem:

For every integer $n$ , let $S(n)$ be the sum of the digits of $n$ (in base ten). What is the smallest integer $N$ for which $S(S(N)) \geq 10$ ?

Solution

Solution:

The answer is $199$ . Indeed if $n$ had $1$ or $2$ digits, $n \leq 99$ , and then $S(n) \leq 18$ and $S(S(n)) \leq 9$ , so $n$ has at least three digits. Observe that for $n \leq 198$ , still $S(n) \leq 18$ . Instead if $n=199$ , $S(n)=19$ and $S(S(n))=10$ , as sought.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.