To break the long hiatus, I’m going to solve a short combinatorics problem extracted from “A Walk Through Combinatorics”:
Find all positive integers such that .
As is the smallest of the three integers, we know that will be the bigger of the three fractions. The three fractions must add to one and they must be different, so must be equal to . Rewriting the restriction using this newly found knowledge:
must be greater than 2, but it cannot be bigger than 3 because, otherwise,
Then we have and