Question 41
For any positive integer $n$ denote by $a_n$ the number of quadratic functions $f(x) = ax^2 + bx + c$, $a, b, c \in \{1, 2, \dots, n\}$, having only integer roots. Prove that for every $n \ge 4$,
$$
n \lt a_n \lt n^2.
$$
$$
n \lt a_n \lt n^2.
$$