Question 75
Let $p \ge 3$ be a prime, and let $r$ be an integer with $0 \le r \le p-3$. Let $x_1, x_2, \dots, x_{p-1+r}$ be integers satisfying
$$
\sum_{i=1}^{p-1+r} x_i^k \equiv r \pmod{p}
$$
for all $1 \le k \le p-2$. What can the remainders of $x_1, x_2, \dots, x_{p-1+r}$ upon division by $p$ be?
$$
\sum_{i=1}^{p-1+r} x_i^k \equiv r \pmod{p}
$$
for all $1 \le k \le p-2$. What can the remainders of $x_1, x_2, \dots, x_{p-1+r}$ upon division by $p$ be?