Romanian Mathematical Olympiad

We will prove by contradiction that there is no integer between $\frac{1}{2} + \sqrt{n + \frac{1}{2020}}$ and $\frac{1}{2} + \sqrt{n + \frac{1}{2}}$.

Suppose there exists $k \ge 1$ such that $\frac{1}{2} + \sqrt{n + \frac{1}{2}} \ge k > \frac{1}{2} + \sqrt{n + \frac{1}{2020}}$. We obtain $n \ge k^2 - k - \frac{1}{4} = x_k$ and $n < k^2 - k + \frac{126}{505} = y_k$.

Since the only integer between $x_k$ and $y_k$ is $k^2 - k$, we must have $n = k^2 - k = k(k - 1)$ which contradicts $n$ odd.