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Problem 871

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Drifting Subsets

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

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漂移子集

记$f$为从有限集$S$到其本身的函数。若$S$的一个子集$A$满足，并集$A \cup f(A)$的元素数目是$A$的元素数目的两倍，则称$A$为$f$的漂移子集。

记$D(f)$为$f$所有漂移子集的元素数目最大值。

对于正整数$n$，定义$f_n$为从集合$\{0, 1, \dots, n - 1\}$到其本身的函数，将任意元素$x$映射到$x^3 + x + 1 \bmod n$。

已知$D(f_5) = 1$，$D(f_{10}) = 3$。

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

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