2018 IMO Shortlist A6

Let $m, n \geq 2$ 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)

令 $m, n \geq 2$ 为整数。令 $f\left(x_{1}, \ldots, x_{n}\right)$ 为具有实数系数的多项式，使得 $$f\left(x_{1}, \ldots, x_{n}\right)=\left\lfloor\frac{x_{1}+\ldots+x_{n}}{m}\right\rfloor \text { 对于每 } x_{1}, \ldots, x_{n} \in\{0,1, \ldots, m-1\}$$证明$f$的总度数至少为$n$。 (巴西)