Question 167
Prove that there exists a real number $\varepsilon \gt 0$ such that there are infinitely many sequences of integers $0 \lt a_1 \lt a_2 \lt \dots \lt a_{2025}$ satisfying
$$
\text{gcd}(a_1^2 + 1, a_2^2 + 1, \dots, a_{2025}^2 + 1) \gt a_{2025}^{1+\varepsilon}.
$$
$$
\text{gcd}(a_1^2 + 1, a_2^2 + 1, \dots, a_{2025}^2 + 1) \gt a_{2025}^{1+\varepsilon}.
$$