2006/07 AHSMC Part 1 Problem 3

Since $100^4 = 100000000$, $(100, 1)$ is a desired pair. In fact, $m$ can be any positive integer up to $100$, and there is a unique positive integer $n$ such that $m^4 + n = 100000001$ for this particular $m$. Hence the total number of pairs is $100$. The answer is (a).