62nd 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: $$x^2\le 2023\le -46x\Rightarrow -44\le x\le 44\text{ and }x\le -44\Rightarrow x=-44\Rightarrow 46-n=-44\Rightarrow n=90.$$