Romanian Mathematical Olympiad - Final Round

Denote by $c$ the quotient of the division. From the quotient-remainder theorem we obtain $2023 = n \cdot c + 223 - \frac{3}{2} \cdot n$, thus $(2c-3)n = 3600$. (1)

The remainder $223 - \frac{3}{2} \cdot n$ is a positive integer, therefore $n$ is even and $0 \le 223 - \frac{3}{2} \cdot n < n$, whence we deduce that $90 \le n \le 148$. Because $n$ is a square, it follows that $n = 100$ or $n = 144$. If $n = 100$, we deduce from (1) that $2c - 3 = 36$, false, and if $n = 144$, we obtain the solution $c = 14$.