International Mathematics Competition
for University Students
2025

IMC2025: Day 2, Problem 10

Problem 10. For any positive integer \(\displaystyle N\), let \(\displaystyle S_N\) be the number of pairs of integers \(\displaystyle 1\leq a, b\leq N\) such that the number \(\displaystyle (a^2+a)(b^2+b)\) is a perfect square. Prove that the limit

\(\displaystyle \lim_{N\to\infty} \frac{S_N}{N} \)

exists and find its value.

Besfort Shala, University of Bristol

  Solution