Question 28
Let $m, n \ge 2$ be integers and let $f(x_1, \dots, x_n)$ be a polynomial with real coefficients such that for every $x_1, x_2, \dots, x_n \in \{0, 1, \dots, m-1\}$,
$$
f(x_1, \dots, x_n) = \left[ \frac{x_1 + \dots + x_n}{m} \right]
$$
holds. Prove that the degree of $f$ is at least $n$.
$$
f(x_1, \dots, x_n) = \left[ \frac{x_1 + \dots + x_n}{m} \right]
$$
holds. Prove that the degree of $f$ is at least $n$.