Ukrainian National Mathematical Olympiad

Let $x = 46 - n$ and find the corresponding integer values of $x$ that satisfy the condition $-46 \le \frac{2023}{x} \le x$. We consider two cases.

Case 1. $x > 0$. The left inequality holds for all the mentioned values of $x$, and the right one can be rewritten as:
$$2023 \le x^2 \Rightarrow x \ge 45 \Rightarrow 46 - n \ge 45 \Rightarrow n \le 1 \Rightarrow n = 1.$$

Case 2. $x < 0$. The inequality can be rewritten as:
$$x^2 \le 2023 \le -46x \Rightarrow -44 \le x \le 44 \text{ and } x \le -44 \Rightarrow x = -44 \Rightarrow 46 - n = -44 \Rightarrow n = 90.$$