Japan Mathematical Olympiad

Let $n$ be a positive integer such that $n + 10$ and $10n$ are both perfect squares. Then, there exists a positive integer $k$ such that $10n = k^2$. Since $k$ is a multiple of $2$ and $5$, we can write $k = 10\ell$ for some positive integer $\ell$, and we have $n = 10\ell^2$. For $\ell = 1$ or $\ell = 2$, we have $n + 10 = 20$ or $50$, respectively, which are not perfect squares. Therefore, we must have $\ell \ge 3$, which implies $n = 10\ell^2 \ge 10 \cdot 3^2 = 90$.

On the other hand, since we have $90 + 10 = 100 = 10^2$ and $90 \cdot 10 = 900 = 30^2$, $90$ satisfies the condition. Therefore, the answer is $90$.