Question 46
Let $n \gt 1$ be a positive integer and $a_1, a_2, \dots, a_n$ be a sequence of $n$ positive integers. Let
$$
b_i = \left[ \frac{a_1 + a_2 + \dots + a_{i-1} + a_{i+1} + \dots + a_n}{n-1} \right], \quad 1 \leq i \leq n.
$$
Let $f$ be a mapping such that $f(a_1, a_2, \dots, a_n) = (b_1, b_2, \dots, b_n)$.
a) Let the function $g: \mathbb{N} \rightarrow \mathbb{N}$ be defined such that $g(1)$ is the number of different elements in the sequence $f(a_1, a_2, \dots, a_n)$ and $g(m)$ is the number of different elements in the sequence $f^m(a_1, a_2, \dots, a_n) = f(f^{m-1}(a_1, a_2, \dots, a_n)),\ m \gt 1$. Prove that there is a positive integer $k_0$ such that for $m \geq k_0$ the function $g(m)$ is periodic.
b) Prove that
$$
\sum_{m=1}^{k} \frac{g(m)}{m(m+1)} \lt C
$$
for any positive integer $k$, where the constant $C$ does not depend on $k$.
$$
b_i = \left[ \frac{a_1 + a_2 + \dots + a_{i-1} + a_{i+1} + \dots + a_n}{n-1} \right], \quad 1 \leq i \leq n.
$$
Let $f$ be a mapping such that $f(a_1, a_2, \dots, a_n) = (b_1, b_2, \dots, b_n)$.
a) Let the function $g: \mathbb{N} \rightarrow \mathbb{N}$ be defined such that $g(1)$ is the number of different elements in the sequence $f(a_1, a_2, \dots, a_n)$ and $g(m)$ is the number of different elements in the sequence $f^m(a_1, a_2, \dots, a_n) = f(f^{m-1}(a_1, a_2, \dots, a_n)),\ m \gt 1$. Prove that there is a positive integer $k_0$ such that for $m \geq k_0$ the function $g(m)$ is periodic.
b) Prove that
$$
\sum_{m=1}^{k} \frac{g(m)}{m(m+1)} \lt C
$$
for any positive integer $k$, where the constant $C$ does not depend on $k$.