Problem 1888

Let $p\ge 3$ be a prime number and $0\le r\le p-3$. Let $x_1, x_2, \ldots, x_{p-1+r}$ be integer numbers satisfying $\sum_{j=1}^{p-1+r}x_j^k\equiv r~ \textrm{(mod}\ p\textrm{)}$ for all $1\le k\le p-2$.