Problem Archive

Let $f$ be a function from a finite set $S$ to itself. A drifting subset for $f$ is a subset $A$ of $S$ such that the number of elements in the union $A \cup f(A)$ is equal to twice the number of elements of $A$.
We write $D(f)$ for the maximal number of elements among all drifting subsets for $f$.

For a positive integer $n$, define $f_n$ as the function from $\{0, 1, \dots, n - 1\}$ to itself sending $x$ to $x^3 + x + 1 \bmod n$.
You are given $D(f_5) = 1$ and $D(f_{10}) = 3$.

Find $\displaystyle\sum_{i = 1}^{100} D(f_{10^5 + i})$.