IMC 2025 · Problem 10

For any positive integer $N$, let $S_N$ be the number of pairs of integers $1 \le a, b \le N$ such that the number $(a^2 + a)(b^2 + b)$ is a perfect square. Prove that the limit

$$\lim_{N \to \infty} \frac{S_N}{N}$$

exists and find its value.