Question 151
Let $n \geq 10$ be an integer. We write $n$ in the decimal system. Let $S(n)$ be the sum of the digits of $n$. A *stump* of $n$ is a positive integer obtained by removing a number (at least one, but not all) of the digits of $n$ from the right end. For example: $23$ is a stump of $2351$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n = S(n) + 9 \cdot T(n)$.