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Difference between revisions of "2001 APMO Problems/Problem 1"

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For any positive integer <math>n</math>, let <math>S(n)</math> be the sum of digits in the decimal representation of <math>n</math>. Any positive integer obtained by removing one or more digits from the right end of the decimal representation of <math>n</math> is called a stump of <math>n</math>. Let <math>T(n)</math> be the sum of all stumps of <math>n</math>. Prove that <math>n = S(n) + 9T(n)</math>. (For example, if <math>n = 238</math>, we have <math>S(n) = 2+3+8 = 13</math>, and stumps <math>2</math> and <math>23</math>, so <math>T(n) = 2+23 = 25</math>. We verify that <math>238 = 13 + 9(25)</math>.)

Latest revision as of 22:57, 25 October 2022

For any positive integer $n$, let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing one or more digits from the right end of the decimal representation of $n$ is called a stump of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n = S(n) + 9T(n)$. (For example, if $n = 238$, we have $S(n) = 2+3+8 = 13$, and stumps $2$ and $23$, so $T(n) = 2+23 = 25$. We verify that $238 = 13 + 9(25)$.)

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