Question 5
Let $n\ge2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, the supplied source prints
$$S_a=\{b\mid0\le b\le p-1,\ \text{and there is }c\text{ such that }P_a(x)\equiv p\pmod p\}.$$
Show that the expression
$$\frac1{p-1}\sum_{a=1}^{p-1}|S_a|$$
is an integer.
Editorial clarification: the displayed source line has an unused $c$ and a free $x$, and its right-hand residue is printed $p$ rather than $b$. The preceding prose and the correctly stated duplicate, 2023 Open Round2 Question3, establish the intended definition
$$S_a=\{b\in\{0,\ldots,p-1\}:\exists c\in\mathbb Z,\ P_a(c)\equiv b\pmod p\}.$$
The solution explicitly distinguishes the defective printed line from this intended image set.
$$S_a=\{b\mid0\le b\le p-1,\ \text{and there is }c\text{ such that }P_a(x)\equiv p\pmod p\}.$$
Show that the expression
$$\frac1{p-1}\sum_{a=1}^{p-1}|S_a|$$
is an integer.
Editorial clarification: the displayed source line has an unused $c$ and a free $x$, and its right-hand residue is printed $p$ rather than $b$. The preceding prose and the correctly stated duplicate, 2023 Open Round2 Question3, establish the intended definition
$$S_a=\{b\in\{0,\ldots,p-1\}:\exists c\in\mathbb Z,\ P_a(c)\equiv b\pmod p\}.$$
The solution explicitly distinguishes the defective printed line from this intended image set.