DE-MO 2019 Problem 5

We are given two positive integers $p$ and $q$ .

Step by step, a rope of length $1$ is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio $p:q$ at the same time. After an unknown number of such operations, the currently longest pieces have the length $x$ .

Determine in terms of $x$ the number $a(x)$ of different lengths of pieces of rope existing at that time.