Question 113
Determine all permutations $(a_1, a_2, \dots, a_{2022})$ of $(1, 2, \dots, 2022)$ with the following property:
for any two positive integers $m, n$ whose difference is greater than $20^{22}$, the following inequality always holds:
$$
\text{gcd}(m + 1, n + a_1) + \text{gcd}(m + 2, n + a_2) + \dots + \text{gcd}(m + 2021, n + a_{2021}) \lt 2|m - n|.
$$
for any two positive integers $m, n$ whose difference is greater than $20^{22}$, the following inequality always holds:
$$
\text{gcd}(m + 1, n + a_1) + \text{gcd}(m + 2, n + a_2) + \dots + \text{gcd}(m + 2021, n + a_{2021}) \lt 2|m - n|.
$$