Question 20
Let $p_1, p_2, \dots, p_{2025}$ be real numbers, and let $\{a_n^{(1)}\}_{n \ge 0}$, $\{a_n^{(2)}\}_{n \ge 0}$, $\dots$, $\{a_n^{(2025)}\}_{n \ge 0}$ be $2025$ real sequences satisfying:
(1) $a_0^{(i)}$ ($1 \le i \le 2025$) are all zero;
(2) $a_1^{(i)}$ ($1 \le i \le 2025$) are **not** all zero;
(3) For $i = 1, 2, \dots, 2025$ and any positive integer $n$,
$$
a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)} = p_i \cdot a_n^{(i+1)},
$$
where $a_n^{(2026)} = a_n^{(1)}$.
Prove that there exists a positive real number $r$ and infinitely many positive integers $n$ such that
$$
\max \{|a_n^{(1)}|, |a_n^{(2)}|, \dots, |a_n^{(2025)}|\} \gt r.
$$
(1) $a_0^{(i)}$ ($1 \le i \le 2025$) are all zero;
(2) $a_1^{(i)}$ ($1 \le i \le 2025$) are **not** all zero;
(3) For $i = 1, 2, \dots, 2025$ and any positive integer $n$,
$$
a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)} = p_i \cdot a_n^{(i+1)},
$$
where $a_n^{(2026)} = a_n^{(1)}$.
Prove that there exists a positive real number $r$ and infinitely many positive integers $n$ such that
$$
\max \{|a_n^{(1)}|, |a_n^{(2)}|, \dots, |a_n^{(2025)}|\} \gt r.
$$