Nemotron-IMO-Bench 2026 A2

Let $m,n\geq 2$ be integers and let $f\in \mathbb{R}[X_1,...,X_n]$ be a polynomial such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor,\, \forall\, x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.