2022 SSMO Tiebreaker Round Problems

Problem 1

For all positive integers $n,$ let $S(n)$ denote the least positive integer $x$ such that $n+x$ is a palindrome. Find the value of ${\sum_{n=1}^{100}S(n)}.$

Solution

Problem 2

Let $P(x) = x^3 - 7x^2 + 9x - 2$ . If $P(x) = (x - a)^3 + b(x - a) + c$ where $a, b, c$ are real numbers, then the value of $a - b - c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Find $m + n$ .