Question 3
Let $n\ge2$ be a positive integer. For any integer $a$, let $Q_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and $S_a$ be the set
$$S_a=\{b\mid0\le b\le p-1,\ \exists c\in\mathbb Z,\ Q_a(c)\equiv b\pmod p\}.$$
Show that the expression $\dfrac1{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.
$$S_a=\{b\mid0\le b\le p-1,\ \exists c\in\mathbb Z,\ Q_a(c)\equiv b\pmod p\}.$$
Show that the expression $\dfrac1{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.