2001 APMO Problems/Problem 1

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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2001 APMO Problems/Problem 1