ISL 2018 A6

Let $m, n \geqslant 2$ be integers. Let $f\left(x_{1}, \ldots, x_{n}\right)$ be a polynomial with real coefficients such that $$f\left(x_{1}, \ldots, x_{n}\right)=\left\lfloor\frac{x_{1}+\ldots+x_{n}}{m}\right\rfloor \text { for every } x_{1}, \ldots, x_{n} \in\{0,1, \ldots, m-1\}$$ Prove that the total degree of $f$ is at least $n$. (Brazil)