Question 25
Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1 + a_2 + a_3 + a_4.$ There is a polynomial $$
q(x) = c_3x^3 + c_2x^2 + c_1x + c_0
$$ such that $D(m) = q(m)$ for all odd integers $m \ge 5.$ What is $c_1?$
q(x) = c_3x^3 + c_2x^2 + c_1x + c_0
$$ such that $D(m) = q(m)$ for all odd integers $m \ge 5.$ What is $c_1?$