Question 187
Let $m$ and $n$ be two positive integers with $m \ge n \ge 2022$. Let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be $2n$ real numbers. Prove that the numbers of ordered pairs $(i, j)$ ($1 \le i, j \le n$) such that
$$
|a_i + b_j - ij| \le m
$$
is less than or equal to $3n\sqrt{m \log n}$.
$$
|a_i + b_j - ij| \le m
$$
is less than or equal to $3n\sqrt{m \log n}$.